Mean and standard deviation versus median and IQR

WEBVTT
Kind: captions
Language: en

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- [Narrator] So we have
nine students who recently

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graduated from a small school
that has a class size of nine,

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and they wanna figure out
what is the central tendency

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for salaries one year after graduation?

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And they also wanna have a
sense of the spread around

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that central tendency one
year after graduation.

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So they all agree to put in
their salaries into a computer,

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and so these are their salaries.

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They're measured in thousands.

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So one makes 35,000, 50,000,
50,000, 50,000, 56,000,

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two make 60,000, one makes
75,000, and one makes 250,000.

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So she's doing very well for herself,

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and the computer it spits
out a bunch of parameters

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based on this data here.

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So it spits out two typical
measures of central tendency.

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The mean is roughly 76.2.

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The computer would calculate
it by adding up all of these

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numbers, these nine numbers,
and then dividing by nine,

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and the median is 56, and median
is quite easy to calculate.

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You just order the numbers and you take

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the middle number here which is 56.

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Now what I want you to
do is pause this video

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and think about for this data set,

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for this population of
salaries, which measure,

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which measure of central
tendency is a better measure?

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All right, so let's think
about this a little bit.

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I'm gonna plot it on a line here.

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I'm gonna plot my data
so we get a better sense

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and we just don't see them,
so we just don't see things

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as numbers, but we see
where those numbers sit

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relative to each other.

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So let's say this is zero.

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Let's say this is, let's see,
one, two, three, four, five.

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So this would be 250, this
is 50, 100, 150, 200, 200,

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and let's see.

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Let's say if this is 50
than this would be roughly

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40 right here, and I just wanna get rough.

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So this would be about 60,
70, 80, 90, close enough.

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I'm, I could draw this
a little bit neater,

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but, 60, 70, 80, 90.

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Actually, let me just clean
this up a little bit more too.

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This one right over here would be

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a little bit closer to this one.

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Let me just put it right around here.

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So that's 40, and then
this would be 30, 20, 10.

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Okay, that's pretty good.

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So let's plot this data.

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So, one student makes 35,000,
so that is right over there.

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Two make 50,000, or three make 50,000,

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so one, two, and three.

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I'll put it like that.

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One makes 56,000 which would
put them right over here.

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One makes 60,000, or
actually, two make 60,000,

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so it's like that.

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One makes 75,000, so
that's 60, 70, 75,000.

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So it's gonna be right around there,

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and then one makes 250,000.

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So one's salary is all
the way around there,

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and then when we
calculate the mean as 76.2

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as our measure of central tendency,

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76.2 is right over there.

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So is this a good measure
of central tendency?

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Well to me it doesn't feel that good,

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because our measure of central
tendency is higher than all

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of the data points except for
one, and the reason is is that

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you have this one that the,
that our, our data is skewed

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significantly by this
data point at $250,000.

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It is so far from the
rest of the distribution

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from the rest of the data
that it has skewed the mean,

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and this is something
that you see in general.

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If you have data that is skewed,
and especially things like

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salary data where someone might
make, most people are making

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50, 60, $70,000, but someone
might make two million dollars,

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and so that will skew the
average or skew the mean I should

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say, when you add them all
up and divide by the number

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of data points you have.

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In this case, especially when
you have data points that

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would skew the mean,
median is much more robust.

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The median at 56 sits right
over here, which seems to be

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much more indicative for central tendency.

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And think about it.

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Even if you made this instead of 250,000

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if you made this 250,000
thousand, which would be 250

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million dollars, which is
a ginormous amount of money

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to make, it wouldn't, it would
skew the mean incredibly,

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but it actually would not
even change the median,

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because the median, it doesn't matter

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how high this number gets.

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This could be a trillion dollars.

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This could be a quadrillion dollars.

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The median is going to stay the same.

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So the median is much more robust

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if you have a skewed data set.

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Mean makes a little bit more
sense if you have a symmetric

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data set or if you have things
that are, you know, where,

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where things are roughly
above and below the mean,

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or things aren't skewed
incredibly in one direction,

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especially by a handful of data

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points like we have right over here.

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So in this example, the median is a much

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better measure of central tendency.

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And so what about spread?

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Well you might say, well,
Sal you already told us

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that the mean is not so good

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and the standard deviation
is based on the mean.

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You take each of these data
points, find their distance

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from the mean, square that
number, add up those squared

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distances, divide by the
number of data points if we're

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taking the population standard
deviation, and then you,

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and then you, you take the
square root of the whole thing.

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And so since this is based on
the mean, which isn't a good

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measure of central tendency
in this situation, and this,

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this is also going to skew
that standard deviation.

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This is going to be, this is a lot larger

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than if you look at the, the actual,

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if you wanted an indication of the spread.

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Yes, you have this one data
point that's way far away

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from either the mean or
the median depending on how

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you wanna think about it, but
most of the data points seem

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much closer, and so for that situation,

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not only are we using the median,

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but the interquartile range
is once again more robust.

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How do we calculate the
interquartile range?

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Well, you take the median
and then you take the bottom

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group of numbers and
calculate the median of those.

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So that's 50 right over here
and then you take the top

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group of numbers, the
upper group of numbers,

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and the median there is
60 and 75, it's 67.5.

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If this looks unfamiliar
we have many videos

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on interquartile range and calculating

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standard deviation and median and mean.

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This is just a little bit of a review,

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and then the difference
between these two is 17.5,

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and notice, this distance
between these two, this 17.5,

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this isn't going to change,

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even if this is 250 billion dollars.

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So once again, it is both of
these measures are more robust

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when you have a skewed data set.

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So the big take away here is
mean and standard deviation,

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they're not bad if you have
a roughly symmetric data set,

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if you don't have any
significant outliers,

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things that really skew the data set,

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mean and standard deviation
can be quite solid.

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But if you're looking at
something that could get really

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skewed by a handful of data
points median might be,

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median and interquartile range,
median for central tendency,

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interquartile range for spread
around that central tendency,

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and that's why you'll see when
people talk about salaries

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they'll often talk about
median, because you can have

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some skewed salaries,
especially on the up side.

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When we talk about things
like home prices you'll see

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median often measured
more typically than mean,

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because home prices in a
neighborhood, a lot of,

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or in a city, a lot of the
houses might be in the 200,000,

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$300,000 range, but maybe
there's one ginormous mansion

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that is 100 million dollars,
and if you calculated mean

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that would skew and give a
false impression of the average

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or the central tendency
of prices in that city.


Bivariate relationship linearity, strength and direction

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we have here

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is six different scatter plots that show

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the relationship between
different variables.

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So, for example, in this one here,

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in the horizontal axis, we might have

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something like age, and then here

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it could be accident frequency.

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Accident frequency.

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And I'm just making this up.

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And I could just show these data points,

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maybe for some kind of statistical survey,

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that, when the age is this,
whatever number this is,

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maybe this is 20 years old,
this is the accident frequency.

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And it could be a number
of accidents per hundred.

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And that, when the age is 21 years old,

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this is the frequency.

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And so, these data
scientists, or statisticians,

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went and plotted all of
these in this scatter plot.

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This is often known as bivariate data,

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which is a very fancy way of saying,

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hey, you're plotting things that take

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two variables into consideration,

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and you're trying to see whether

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there's a pattern with how they relate.

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And what we're going to do in this video

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is think about, well,
can we try to fit a line,

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does it look like there's a linear

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or non-linear relationship between

00:01:04.933 --> 00:01:07.243
the variables on the different axes?

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How strong is that variable?

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Is it a positive, is it
a negative relationship?

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And then, we'll think about
this idea of outliers.

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So let's just first think about

00:01:15.896 --> 00:01:18.369
whether there's a linear
or non-linear relationship.

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And I'll get my little
ruler tool out here.

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So, this data right over here,

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it looks like I could get a,
I could put a line through it

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that gets pretty close through the data.

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You're not gonna, it's very unlikely

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you're gonna be able to go
through all of the data points,

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but you can try to get a
line, and I'm just doing this.

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There's more numerical, more
precise ways of doing this,

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but I'm just eyeballing
it right over here.

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And it looks like I could plot a line

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that looks something like that,

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that goes roughly through the data.

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So this looks pretty linear.

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And so I would call this
a linear relationship.

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And since, as we increase one variable,

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it looks like the other
variable decreases.

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This is a downward-sloping line.

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I would say this is a negative.

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This is a negative linear relationship.

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But this one looks pretty strong.

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So, because the dots aren't
that far from my line.

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This one gets a little bit further,

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but it's not, there's not
some dots way out there.

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And so, most of 'em are
pretty close to the line.

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So I would call this a negative,

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reasonably strong linear relationship.

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Negative, strong, I'll call it reasonably,

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I'll just say strong,
but reasonably strong,

00:02:28.507 --> 00:02:32.674
linear, linear relationship
between these two variables.

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Now, let's look at this one.

00:02:35.088 --> 00:02:36.298
And pause this video and think about

00:02:36.298 --> 00:02:39.495
what this one would be for you.

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Well, let's see.

00:02:40.333 --> 00:02:42.958
I'll get my ruler tool out again.

00:02:42.958 --> 00:02:44.887
And it looks like I can try to put a line,

00:02:44.887 --> 00:02:46.761
it looks like, generally speaking,

00:02:46.761 --> 00:02:49.447
as one variable increases,
the other variable

00:02:49.447 --> 00:02:52.142
increases as well, so something like this

00:02:52.142 --> 00:02:56.774
goes through the data and
approximates the direction.

00:02:56.774 --> 00:02:57.791
And this looks positive.

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As one variable increases, the other

00:02:59.889 --> 00:03:02.238
variable increases, roughly.

00:03:02.238 --> 00:03:05.055
So this is a positive relationship.

00:03:05.055 --> 00:03:06.213
But this is weak.

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A lot of the data is off,
well off of the line.

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So, positive, weak.

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But I'd say this is still linear.

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It seems that, as we increase one,

00:03:18.210 --> 00:03:20.300
the other one increases
at roughly the same rate,

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although these data points
are all over the place.

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So, I would still call this linear.

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Now, there's also this notion of outliers.

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If I said, hey, this line is trying

00:03:29.355 --> 00:03:32.758
to describe the data,
well, we have some data

00:03:32.758 --> 00:03:35.479
that is fairly off the line.

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So, for example, even though we're saying

00:03:37.552 --> 00:03:40.315
it's a positive, weak,
linear relationship,

00:03:40.315 --> 00:03:42.656
this one over here is reasonably high

00:03:42.656 --> 00:03:44.765
on the vertical variable, but it's low

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on the horizontal variable.

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And so, this one right
over here is an outlier.

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It's quite far away from the line.

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You could view that as an outlier.

00:03:52.614 --> 00:03:54.307
And this is a little bit subjective.

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Outliers, well, what looks pretty

00:03:55.852 --> 00:03:57.942
far from the rest of the data?

00:03:57.942 --> 00:04:00.618
This could also be an outlier.

00:04:00.618 --> 00:04:02.272
Let me label these.

00:04:02.272 --> 00:04:03.105
Outlier.

00:04:04.994 --> 00:04:05.973
Now, pause the video and see if

00:04:05.973 --> 00:04:06.806
you can think about this one.

00:04:06.806 --> 00:04:08.712
Is this positive or
negative, is it linear,

00:04:08.712 --> 00:04:11.495
non-linear, is it strong or weak?

00:04:11.495 --> 00:04:13.509
I'll get my ruler tool out here.

00:04:13.509 --> 00:04:15.632
So, this goes here.

00:04:15.632 --> 00:04:19.677
It seems like I can fit a
line pretty well to this.

00:04:19.677 --> 00:04:23.911
So, I could fit, maybe
I'll do the line in purple.

00:04:23.911 --> 00:04:26.495
I could fit a line that looks like that.

00:04:26.495 --> 00:04:28.410
And so, this one looks like it's positive.

00:04:28.410 --> 00:04:30.605
As one variable increases,
the other one does,

00:04:30.605 --> 00:04:32.148
for these data points.

00:04:32.148 --> 00:04:33.774
So it's a positive.

00:04:33.774 --> 00:04:35.383
I'd say this was pretty strong.

00:04:35.383 --> 00:04:37.666
The dots are pretty
close to the line there.

00:04:37.666 --> 00:04:38.833
It really does look like a little bit

00:04:38.833 --> 00:04:40.968
of a fat line, if you
just look at the dots.

00:04:40.968 --> 00:04:42.718
So, positive, strong,

00:04:44.481 --> 00:04:46.814
linear, linear relationship.

00:04:48.256 --> 00:04:52.092
And none of these data points
are really strong outliers.

00:04:52.092 --> 00:04:53.617
This one's a little bit further out.

00:04:53.617 --> 00:04:55.303
But they're all pretty close to the line,

00:04:55.303 --> 00:04:58.927
and seem to describe that trend roughly.

00:04:58.927 --> 00:05:02.680
All right, now, let's look
at this data right over here.

00:05:02.680 --> 00:05:06.317
So, let me get my line tool out again.

00:05:06.317 --> 00:05:09.578
So, it looks like I can fit a line.

00:05:09.578 --> 00:05:11.800
So it looks, and it looks like
it's a positive relationship.

00:05:11.800 --> 00:05:13.296
The line would be upward sloping.

00:05:13.296 --> 00:05:15.123
It would look something like this.

00:05:15.123 --> 00:05:17.564
And, once again, I'm eyeballing it.

00:05:17.564 --> 00:05:19.307
You can use computers and other methods

00:05:19.307 --> 00:05:21.669
to actually find a more precise line

00:05:21.669 --> 00:05:23.491
that minimizes the collective distance

00:05:23.491 --> 00:05:26.487
to all of the points, but it looks like

00:05:26.487 --> 00:05:29.713
there is a positive, but I would say,

00:05:29.713 --> 00:05:32.040
this one is a weak linear relationship,

00:05:32.040 --> 00:05:34.703
'cause we have a lot of points
that are far off the line.

00:05:34.703 --> 00:05:36.446
So, not so strong.

00:05:36.446 --> 00:05:38.904
So, I would call this a positive,

00:05:38.904 --> 00:05:41.821
weak, linear relationship.

00:05:41.821 --> 00:05:43.781
And there's a lot of outliers here.

00:05:43.781 --> 00:05:48.328
This one over here is
pretty far, pretty far out.

00:05:48.328 --> 00:05:49.957
Now, let's look at this one.

00:05:49.957 --> 00:05:50.955
Pause this video and think about,

00:05:50.955 --> 00:05:52.839
is it positive or negative,
is strong or weak?

00:05:52.839 --> 00:05:54.842
Is this linear or non-linear?

00:05:54.842 --> 00:05:55.821
Well, the first thing we wanna do

00:05:55.821 --> 00:05:57.390
is let's think about it
with linear or non-linear.

00:05:57.390 --> 00:06:00.281
I could try to put a line on it.

00:06:00.281 --> 00:06:02.024
But if I try to put a line on it,

00:06:02.024 --> 00:06:03.845
it's actually quite difficult.

00:06:03.845 --> 00:06:05.381
If I try to do a line like this,

00:06:05.381 --> 00:06:06.214
you'll notice everything is kind

00:06:06.214 --> 00:06:08.554
of bending away from the line.

00:06:08.554 --> 00:06:11.163
It looks like, generally,
as one variable increases,

00:06:11.163 --> 00:06:12.686
the other variable decreases, but they're

00:06:12.686 --> 00:06:15.065
not doing it in a linear fashion.

00:06:15.065 --> 00:06:18.071
It looks like there's some
other type of curve at play.

00:06:18.071 --> 00:06:20.267
So, I could try to do a fancier curve

00:06:20.267 --> 00:06:21.634
that looks something like this,

00:06:21.634 --> 00:06:24.683
and this seems to fit
the data a lot better.

00:06:24.683 --> 00:06:28.820
So this one, I would
describe as non-linear.

00:06:28.820 --> 00:06:30.607
And it is a negative relationship.

00:06:30.607 --> 00:06:33.968
As one variable increases,
the other variable decreases.

00:06:33.968 --> 00:06:37.272
So, this is a negative, I would say,

00:06:37.272 --> 00:06:40.204
reasonably strong non-linear relationship.

00:06:40.204 --> 00:06:42.104
Pretty strong.

00:06:42.104 --> 00:06:43.057
Pretty strong.

00:06:43.057 --> 00:06:44.838
And once again, this is subjective.

00:06:44.838 --> 00:06:48.110
So, I'll say negative, reasonably strong,

00:06:48.110 --> 00:06:50.936
non-linear relationship.

00:06:50.936 --> 00:06:53.694
And maybe you could call
this one an outlier,

00:06:53.694 --> 00:06:55.772
but it's not that far,
and I might even be able

00:06:55.772 --> 00:06:57.718
to fit a curve that gets a
little bit closer to that.

00:06:57.718 --> 00:06:59.675
And once again, I'm eyeballing this.

00:06:59.675 --> 00:07:01.737
Now let's do this last one.

00:07:01.737 --> 00:07:06.385
And so, this one looks like a
negative linear relationship

00:07:06.385 --> 00:07:10.564
to me, a fairly strong
negative linear relationship,

00:07:10.564 --> 00:07:12.675
although there are some outliers.

00:07:12.675 --> 00:07:15.479
So, let me draw this line.

00:07:15.479 --> 00:07:18.230
So that seems to fit the data pretty good.

00:07:18.230 --> 00:07:21.647
So this is a negative, reasonably strong,

00:07:22.529 --> 00:07:26.172
reasonably strong linear relationship.

00:07:26.172 --> 00:07:28.677
But these are very clear outliers.

00:07:28.677 --> 00:07:31.369
These are well away from the data,

00:07:31.369 --> 00:07:34.133
or from the cluster of where
most of the points are.

00:07:34.133 --> 00:07:35.941
So, with some significant, with at least

00:07:35.941 --> 00:07:38.886
these two significant outliers here.

00:07:38.886 --> 00:07:40.538
So hopefully this makes
you a little bit familiar

00:07:40.538 --> 00:07:42.349
with some of this terminology, and it's

00:07:42.349 --> 00:07:44.999
important to keep in mind, this
is a little bit subjective.

00:07:44.999 --> 00:07:48.786
There'll be some cases that
are more obvious than others.

00:07:48.786 --> 00:07:50.661
And oftentimes, you
wanna make a comparison,

00:07:50.661 --> 00:07:52.250
that this is a stronger linear,

00:07:52.250 --> 00:07:54.456
positive linear relationship
than this one is,

00:07:54.456 --> 00:07:55.539
right over here, 'cause you can see,

00:07:55.539 --> 00:07:58.483
most of the data is closer to the line.

00:07:58.483 --> 00:08:03.050
This one is, for sure, this is
more non-linear than linear.

00:08:03.050 --> 00:08:05.733
It depends how you wanna describe,

00:08:05.733 --> 00:08:07.697
oftentimes, making a comparison,

00:08:07.697 --> 00:08:11.864
or making a subjective call
on how to describe the data.


Counting faces and edges of 3D shapes

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:04.140
- [Instructor] How many faces
does the following shape have?

00:00:04.140 --> 00:00:07.200
Pause this video and see
if you can figure that out.

00:00:07.200 --> 00:00:08.640
All right, I'm assuming you paused

00:00:08.640 --> 00:00:10.380
and I'll see if we can
work through it together.

00:00:10.380 --> 00:00:13.530
And I'm gonna actually
try to color the faces.

00:00:13.530 --> 00:00:17.220
So we have this face over, whoops.

00:00:17.220 --> 00:00:20.520
Let me do it in this other tool.

00:00:20.520 --> 00:00:24.750
So you have this face
over here in the back.

00:00:24.750 --> 00:00:27.000
So that's one face.

00:00:27.000 --> 00:00:28.590
So that's one.

00:00:28.590 --> 00:00:32.700
Then you have this face right over here.

00:00:32.700 --> 00:00:34.770
Also in the back.

00:00:34.770 --> 00:00:36.390
The only way we can see this is

00:00:36.390 --> 00:00:39.420
because they've drawn it
so that it is transparent.

00:00:39.420 --> 00:00:41.193
So that is the second face.

00:00:42.450 --> 00:00:45.090
Now you have this triangular face on top.

00:00:45.090 --> 00:00:46.443
So let me color that in.

00:00:47.370 --> 00:00:52.370
So you have this triangular face on top.

00:00:53.100 --> 00:00:55.953
So that that's going to be our third face.

00:00:56.850 --> 00:00:58.170
Third face.

00:00:58.170 --> 00:01:00.690
And then you have this
triangular face on the bottom.

00:01:00.690 --> 00:01:02.403
That's gonna be our fourth face.

00:01:03.963 --> 00:01:05.610
And that's going to be our fourth face.

00:01:05.610 --> 00:01:07.980
And then the key question is, are we done?

00:01:07.980 --> 00:01:11.400
It looks like I've colored
all the ones that I can see,

00:01:11.400 --> 00:01:13.050
but there's one a little bit tricky here.

00:01:13.050 --> 00:01:15.780
There's the one that we are
actually seeing through,

00:01:15.780 --> 00:01:20.340
there's the face,
there's, let pick a color.

00:01:20.340 --> 00:01:24.390
There's the face out front
that we can see through

00:01:24.390 --> 00:01:26.700
so that we can see faces
one, two, and four.

00:01:26.700 --> 00:01:30.150
So that's actually going
to be our fifth face.

00:01:30.150 --> 00:01:32.220
The way they've drawn it, it's
like it's made out of glass.

00:01:32.220 --> 00:01:34.470
So we can see faces one, two, and four.

00:01:34.470 --> 00:01:36.510
But that is our fifth face.

00:01:36.510 --> 00:01:40.470
And so this thing has five faces.

00:01:40.470 --> 00:01:42.000
All right, let's do another example.

00:01:42.000 --> 00:01:45.630
But instead of faces,
let's gonna think about,

00:01:45.630 --> 00:01:48.210
we're gonna think about edges.

00:01:48.210 --> 00:01:51.870
So how many edges does
the following shape have?

00:01:51.870 --> 00:01:55.083
Pause the video and see if
you can work through this.

00:01:56.310 --> 00:01:58.230
Okay, let's work through
this together now.

00:01:58.230 --> 00:02:00.540
And I'm just gonna count these edges.

00:02:00.540 --> 00:02:03.570
So the edges are where two faces meet.

00:02:03.570 --> 00:02:07.290
So this is an edge right over there.

00:02:07.290 --> 00:02:09.060
So that's one edge.

00:02:09.060 --> 00:02:11.100
There's an edge back here,

00:02:11.100 --> 00:02:12.720
we can see 'cause it's transparent.

00:02:12.720 --> 00:02:15.600
That is our second edge.

00:02:15.600 --> 00:02:17.010
We have this one over here.

00:02:17.010 --> 00:02:20.370
That is our third edge.

00:02:20.370 --> 00:02:23.610
We have this one over there.

00:02:23.610 --> 00:02:26.010
That is our fourth edge.

00:02:26.010 --> 00:02:28.560
Then we have this one over here.

00:02:28.560 --> 00:02:31.080
That is our fifth edge.

00:02:31.080 --> 00:02:34.170
Now we have this here.

00:02:34.170 --> 00:02:36.720
This is our sixth edge.

00:02:36.720 --> 00:02:37.802
Let's see.

00:02:37.802 --> 00:02:41.730
All we have left is this one,
which is edge number seven.

00:02:41.730 --> 00:02:44.730
And then last but not
least, this edge over here,

00:02:44.730 --> 00:02:46.110
which is edge number eight.

00:02:46.110 --> 00:02:48.450
And I actually found this
very valuable to color 'em in

00:02:48.450 --> 00:02:50.310
to make sure that I wasn't missing an edge

00:02:50.310 --> 00:02:51.390
or double counting an edge.

00:02:51.390 --> 00:02:55.713
So this thing has eight, eight edges.

00:02:56.640 --> 00:02:58.830
Actually, if you're curious,
just for extra practice,

00:02:58.830 --> 00:03:00.660
how many faces does this have?

00:03:00.660 --> 00:03:03.000
Well, we can count those as well.

00:03:03.000 --> 00:03:08.000
This has one face back there,
another face back there.

00:03:08.010 --> 00:03:09.870
So that's two faces.

00:03:09.870 --> 00:03:12.840
And then you have a third
face, which is the base,

00:03:12.840 --> 00:03:15.090
this rectangular face.

00:03:15.090 --> 00:03:16.710
So that's three faces.

00:03:16.710 --> 00:03:19.680
And then you have your
two faces out front.

00:03:19.680 --> 00:03:24.680
You have this face that
we're seeing through,

00:03:24.900 --> 00:03:27.630
and then we have that face
that we're seeing through.

00:03:27.630 --> 00:03:30.060
So even though they're asking
edges just for practice,

00:03:30.060 --> 00:03:32.820
we figured out that this
thing has five faces.

00:03:32.820 --> 00:03:35.250
One for the square base,

00:03:35.250 --> 00:03:39.370
and then four triangular
faces for these sides

00:03:40.903 --> 00:03:42.060
to make this square pyramid.

00:03:42.060 --> 00:03:45.183
So it has eight edges and five faces.


Finding height of a parallelogram

WEBVTT
Kind: captions
Language: en

00:00:00.090 --> 00:00:02.190
- [Lecturer] The parallelogram shown below

00:00:02.190 --> 00:00:06.690
has an area of 24 units
squared, or square units.

00:00:06.690 --> 00:00:08.700
Find the missing height.

00:00:08.700 --> 00:00:10.950
So, here's the parallelogram.

00:00:10.950 --> 00:00:14.970
This side has length six,
this side has length five.

00:00:14.970 --> 00:00:17.250
And we wanna find the missing height.

00:00:17.250 --> 00:00:19.350
And they gave us the area.

00:00:19.350 --> 00:00:21.000
So, pause this video and see if

00:00:21.000 --> 00:00:22.700
you can figure it out on your own.

00:00:23.580 --> 00:00:25.860
Well the key to solving this,

00:00:25.860 --> 00:00:30.860
is to realize how the area
relates to a base and height.

00:00:31.080 --> 00:00:33.180
An area of a parallelogram,

00:00:33.180 --> 00:00:37.503
area of a parallelogram is
going to be equal to the base,

00:00:38.340 --> 00:00:40.323
times the height.

00:00:41.250 --> 00:00:43.650
Now, what's the base in this scenario?

00:00:43.650 --> 00:00:45.600
Well, you could view this length,

00:00:45.600 --> 00:00:48.180
right over here as the base.

00:00:48.180 --> 00:00:51.240
Which is also going to be
the same as that length.

00:00:51.240 --> 00:00:54.630
In a parallelogram, opposite
sides have the same length.

00:00:54.630 --> 00:00:57.360
The parallel sides have the same length.

00:00:57.360 --> 00:00:58.863
So this going to be six.

00:00:59.753 --> 00:01:01.500
So our base is going to be six.

00:01:01.500 --> 00:01:03.840
And our height is what
we wanna figure out.

00:01:03.840 --> 00:01:07.050
So that, I will just
write as, this is the H

00:01:07.050 --> 00:01:08.700
that we're trying to figure out.

00:01:08.700 --> 00:01:10.200
And we know the area.

00:01:10.200 --> 00:01:12.543
The area is 24 square units.

00:01:13.500 --> 00:01:18.333
So there we go, we have
24 is equal to six.

00:01:19.494 --> 00:01:23.583
Is equal to...six times the height.

00:01:25.623 --> 00:01:29.250
So six times what, is equal to 24?

00:01:29.250 --> 00:01:32.243
Well six times four is equal to 24.

00:01:33.497 --> 00:01:34.330
So we know that H is going
to be equal to four units.

00:01:37.720 --> 00:01:39.803
H is equal to four units.


Surface area word problem example

WEBVTT
Kind: captions
Language: en

00:00:01.080 --> 00:00:03.810
- [Instructor] Akira receives
a prize at a science fair

00:00:03.810 --> 00:00:06.690
for having the most informative project.

00:00:06.690 --> 00:00:10.020
Her trophy is in the
shape of a square pyramid

00:00:10.020 --> 00:00:12.960
and is covered in shiny gold foil.

00:00:12.960 --> 00:00:15.270
So this is what her trophy looks like.

00:00:15.270 --> 00:00:18.660
How much gold foil did it
take to cover the trophy,

00:00:18.660 --> 00:00:20.970
including the bottom?

00:00:20.970 --> 00:00:22.620
And so they give us some dimensions

00:00:22.620 --> 00:00:23.940
and we want how much gold foil,

00:00:23.940 --> 00:00:25.260
and it's an inches squared,

00:00:25.260 --> 00:00:27.180
so it's really going to be an area.

00:00:27.180 --> 00:00:29.370
So pause this video and see
if you can figure that out.

00:00:29.370 --> 00:00:32.733
How much gold foil did it
take to cover the trophy?

00:00:33.900 --> 00:00:36.030
All right, now let's work
through this together.

00:00:36.030 --> 00:00:37.830
And so essentially, what they're asking

00:00:37.830 --> 00:00:41.910
is what is the surface area
of this square pyramid?

00:00:41.910 --> 00:00:44.010
And we're gonna include the base,

00:00:44.010 --> 00:00:46.530
'cause that surface area is how much,

00:00:46.530 --> 00:00:50.010
it's the area of the
gold foil that is needed.

00:00:50.010 --> 00:00:51.840
Now, sometimes, some of you might be able

00:00:51.840 --> 00:00:54.690
to think about this just
by looking at this figure,

00:00:54.690 --> 00:00:56.880
but just to make sure
we don't miss any area,

00:00:56.880 --> 00:00:59.370
I'm gonna open up this square pyramid

00:00:59.370 --> 00:01:01.380
and think about it in two dimensions.

00:01:01.380 --> 00:01:04.260
So what we're gonna do is
imagine if I were to unleash,

00:01:04.260 --> 00:01:08.010
or if I were to cut the top
and, let me do this in red,

00:01:08.010 --> 00:01:12.990
if I were to cut this edge,
if I were to cut this edge,

00:01:14.280 --> 00:01:16.590
if I were to cut that edge,

00:01:16.590 --> 00:01:20.040
and that edge, so the edges that connect

00:01:20.040 --> 00:01:24.300
the triangular sides, and if
I were to just open it all up,

00:01:24.300 --> 00:01:25.860
what would this look like?

00:01:25.860 --> 00:01:27.630
So if I were to open it all up.

00:01:27.630 --> 00:01:29.850
Well at the bottom, you
would have your square base.

00:01:29.850 --> 00:01:33.633
Let me color that in. So
you have your square base.

00:01:34.890 --> 00:01:36.273
So let me draw that.

00:01:37.440 --> 00:01:39.243
So you have your square base.

00:01:40.530 --> 00:01:42.180
This is gonna be a rough drawing.

00:01:43.110 --> 00:01:46.890
And what are the dimensions
there? It's three by three.

00:01:46.890 --> 00:01:48.390
We know this is a square pyramid,

00:01:48.390 --> 00:01:50.550
so the base, all the
sides are the same length.

00:01:50.550 --> 00:01:52.710
They give us one side, but
then if this is three inches,

00:01:52.710 --> 00:01:54.780
and this is gonna be three inches as well.

00:01:54.780 --> 00:01:57.120
And let me color that same
color, just so we recognize

00:01:57.120 --> 00:02:00.090
that we're talking about this same base.

00:02:00.090 --> 00:02:03.210
And if we open up the triangular faces,

00:02:03.210 --> 00:02:04.830
what's it going to look like?

00:02:04.830 --> 00:02:07.890
Well, this is going to look like this.

00:02:07.890 --> 00:02:11.100
This is a rough hand drawing,
but hopefully it makes sense.

00:02:11.100 --> 00:02:13.560
This is going to look like this.

00:02:13.560 --> 00:02:15.420
And each of these triangular faces,

00:02:15.420 --> 00:02:17.823
they all have the exact same area,

00:02:19.560 --> 00:02:20.520
and the reason why I know that,

00:02:20.520 --> 00:02:22.230
they all have the same base, three,

00:02:22.230 --> 00:02:24.720
and they all have the
same height, six inches,

00:02:24.720 --> 00:02:26.910
but I'll draw that in a second.

00:02:26.910 --> 00:02:31.440
So they all look something like
this, just hand drawing it.

00:02:31.440 --> 00:02:33.990
And all of their heights,

00:02:33.990 --> 00:02:35.790
all of their heights are six inches.

00:02:35.790 --> 00:02:38.310
So this right over here is six inches.

00:02:38.310 --> 00:02:41.040
This over here is six inches.

00:02:41.040 --> 00:02:43.530
This over here is six inches.

00:02:43.530 --> 00:02:46.110
And this over here is six inches.

00:02:46.110 --> 00:02:48.720
So to figure out how
much gold foil we need,

00:02:48.720 --> 00:02:50.040
we're trying to figure
out the surface area,

00:02:50.040 --> 00:02:52.020
which is really just
gonna be the combined area

00:02:52.020 --> 00:02:53.580
of these figures.

00:02:53.580 --> 00:02:56.040
Well, the area of this central square

00:02:56.040 --> 00:02:57.660
is pretty easy to figure out.

00:02:57.660 --> 00:02:59.430
It's three inches by three inches,

00:02:59.430 --> 00:03:04.430
so it would be nine inches,
nine inches squared.

00:03:05.100 --> 00:03:07.230
Now, what are the area of the triangles?

00:03:07.230 --> 00:03:09.240
Well, we could figure out the
area of one of the triangles

00:03:09.240 --> 00:03:12.390
and then multiply by four
since there are four triangles.

00:03:12.390 --> 00:03:15.450
So the area of this
triangle right over here,

00:03:15.450 --> 00:03:18.840
it's gonna be one half times our base,

00:03:18.840 --> 00:03:23.840
which is three times three
times our height, which is six.

00:03:25.650 --> 00:03:27.690
Let's see, one half times three times six.

00:03:27.690 --> 00:03:31.710
That's one half times 18,
which is equal to nine,

00:03:31.710 --> 00:03:36.090
nine square inches or nine inches squared.

00:03:36.090 --> 00:03:38.070
So what's gonna be our total area?

00:03:38.070 --> 00:03:41.430
Well, you have the area
of your square base

00:03:41.430 --> 00:03:44.160
plus you have the four sides,

00:03:44.160 --> 00:03:48.360
which each have an area of
nine, so I could write it out.

00:03:48.360 --> 00:03:51.000
I could write four times
nine, or I could write nine.

00:03:51.000 --> 00:03:53.730
Do that black color, or I could write

00:03:53.730 --> 00:03:57.570
nine plus nine plus nine plus nine.

00:03:57.570 --> 00:03:59.430
And just to remind ourselves,

00:03:59.430 --> 00:04:01.110
this is, that right over there,

00:04:01.110 --> 00:04:06.110
is the area of one triangular face.

00:04:08.010 --> 00:04:09.573
Triangular face.

00:04:12.810 --> 00:04:15.750
So this is all of the triangular faces.

00:04:15.750 --> 00:04:20.700
Triangular faces, and of
course ,we have to add that

00:04:20.700 --> 00:04:22.590
to the area of their square base.

00:04:22.590 --> 00:04:24.870
So this is nine plus nine times four.

00:04:24.870 --> 00:04:26.760
You could view this as nine times five,

00:04:26.760 --> 00:04:29.790
which is gonna be 45 square inches,

00:04:29.790 --> 00:04:32.043
nine plus nine plus nine plus nine.


Example of shapes on a coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:03.300
- [Instructor] So we're
told here the four corners

00:00:03.300 --> 00:00:05.017
of a rectangle are located at the points

00:00:05.017 --> 00:00:09.690
one comma one, one comma
six, nine comma six,

00:00:09.690 --> 00:00:11.460
and nine comma one.

00:00:11.460 --> 00:00:12.960
Plot the four corners

00:00:12.960 --> 00:00:15.600
of the rectangle on the
coordinate plane below.

00:00:15.600 --> 00:00:17.400
And they gave us these four points

00:00:17.400 --> 00:00:20.130
and we can move 'em around with our mouse

00:00:20.130 --> 00:00:21.960
or our finger depending on what type

00:00:21.960 --> 00:00:23.850
of a computer we are using.

00:00:23.850 --> 00:00:25.620
And so let's just go point by point

00:00:25.620 --> 00:00:28.350
and plot the green points at those points.

00:00:28.350 --> 00:00:30.330
So the first one is one comma one.

00:00:30.330 --> 00:00:33.840
And remember the first
coordinate is our X coordinate.

00:00:33.840 --> 00:00:35.730
The second coordinate is our Y coordinate.

00:00:35.730 --> 00:00:38.085
So the first coordinate
tells us how far do we move

00:00:38.085 --> 00:00:40.050
to the right of the origin.

00:00:40.050 --> 00:00:41.070
So it's one,

00:00:41.070 --> 00:00:42.360
and then the second coordinate,

00:00:42.360 --> 00:00:44.130
the Y coordinate, tells us how far

00:00:44.130 --> 00:00:45.870
to move up from the origin.

00:00:45.870 --> 00:00:47.700
So that's also a one.

00:00:47.700 --> 00:00:50.130
So one comma one.

00:00:50.130 --> 00:00:53.190
The next point is one comma six.

00:00:53.190 --> 00:00:54.930
X coordinate is one.

00:00:54.930 --> 00:00:57.030
So we move one to the right of the origin,

00:00:57.030 --> 00:00:58.830
and then the Y coordinate is six.

00:00:58.830 --> 00:01:01.590
So we move six up

00:01:01.590 --> 00:01:04.920
and notice it's at the
intersection of the line,

00:01:04.920 --> 00:01:07.770
or it's at the intersection
of when Y equals six

00:01:07.770 --> 00:01:09.240
and X equals one.

00:01:09.240 --> 00:01:13.710
This is one comma six, all right.

00:01:13.710 --> 00:01:16.590
Now we have nine comma six.

00:01:16.590 --> 00:01:19.020
So let's see if we take our,

00:01:19.020 --> 00:01:21.360
if we have X equals nine right over there

00:01:21.360 --> 00:01:23.190
and Y is equal to six.

00:01:23.190 --> 00:01:25.020
So we'd go up six.

00:01:25.020 --> 00:01:27.660
So notice Y is now equal to six.

00:01:27.660 --> 00:01:31.710
And we have one last point
to plot, nine comma one.

00:01:31.710 --> 00:01:35.640
So when X is nine, Y is one.

00:01:35.640 --> 00:01:39.660
We go nine to the right, or
we're right above X equals nine.

00:01:39.660 --> 00:01:41.430
And then we go up one.

00:01:41.430 --> 00:01:42.720
This is nine comma one.

00:01:42.720 --> 00:01:43.650
And there you have it.

00:01:43.650 --> 00:01:46.830
We have the four corners of our rectangle.

00:01:46.830 --> 00:01:49.770
Then they say what is the
height of the rectangle?

00:01:49.770 --> 00:01:52.620
Well, if you imagine a
rectangle right over here,

00:01:52.620 --> 00:01:54.780
the height would be the
distance between that point

00:01:54.780 --> 00:01:56.370
and this point, or the distance between

00:01:56.370 --> 00:01:58.260
that point and that point.

00:01:58.260 --> 00:02:01.140
And so what is the distance
between these points?

00:02:01.140 --> 00:02:02.100
Let's see.

00:02:02.100 --> 00:02:04.230
They both have the same X coordinate

00:02:04.230 --> 00:02:06.060
and this one is at Y equals six.

00:02:06.060 --> 00:02:08.310
This is at Y equals one.

00:02:08.310 --> 00:02:11.010
So this is five higher than this one.

00:02:11.010 --> 00:02:13.140
So the height is five.

00:02:13.140 --> 00:02:14.490
And we could also count it.

00:02:14.490 --> 00:02:19.403
You could see one, two, three, four, five.


Relating circumference and area

WEBVTT
Kind: captions
Language: en

00:00:00.094 --> 00:00:01.454
- [Instructor] So we have a circle here,

00:00:01.454 --> 00:00:03.527
let's say that we know
that its circumference

00:00:03.527 --> 00:00:06.944
is equal to six pi, I'll write it, units.

00:00:08.531 --> 00:00:10.795
Whatever our units happen to be.

00:00:10.795 --> 00:00:12.211
Let's see if we can figure out,

00:00:12.211 --> 00:00:15.967
given that its circumference
is 6 pi of these units,

00:00:15.967 --> 00:00:18.689
what is the area going to be equal to?

00:00:18.689 --> 00:00:21.288
Pause this video and see
if you can figure it out

00:00:21.288 --> 00:00:22.724
on your own.

00:00:22.724 --> 00:00:24.581
And first, think about
if you could figure out

00:00:24.581 --> 00:00:26.909
the area for this particular circle,

00:00:26.909 --> 00:00:28.175
and then let's see if we can come up

00:00:28.175 --> 00:00:30.518
with a formula for
given any circumference,

00:00:30.518 --> 00:00:33.903
can we figure out the area and vice versa.

00:00:33.903 --> 00:00:35.623
Alright now let's work
through this together.

00:00:35.623 --> 00:00:36.591
The key here is to realize

00:00:36.591 --> 00:00:39.407
that from circumference
you can figure out radius

00:00:39.407 --> 00:00:42.327
and then from radius
you can figure out area.

00:00:42.327 --> 00:00:46.721
So we know that circumference,
which is six pi,

00:00:46.721 --> 00:00:50.888
so we know six pi is equal
to two pi times r, radius.

00:00:52.573 --> 00:00:55.735
And so what is the radius going to be?

00:00:55.735 --> 00:00:58.750
So the radius, we're
talking about that distance,

00:00:58.750 --> 00:01:02.271
well we can divide both sides by two pi,

00:01:02.271 --> 00:01:03.751
so let's do that.

00:01:03.751 --> 00:01:06.598
If we divide both sides by two pi,

00:01:06.598 --> 00:01:09.181
to solve for r, what are we left with?

00:01:09.181 --> 00:01:12.014
Well we have an r on the right hand side,

00:01:12.014 --> 00:01:15.294
we have r is equal to pi divided by pi,

00:01:15.294 --> 00:01:16.501
that's just one.

00:01:16.501 --> 00:01:18.985
Six divided by two is three.

00:01:18.985 --> 00:01:21.888
So we get that our
radius, right over here,

00:01:21.888 --> 00:01:23.888
is equal to three units.

00:01:25.093 --> 00:01:26.927
And then we can use the fact

00:01:26.927 --> 00:01:30.317
that are is equal to pi, r squared,

00:01:30.317 --> 00:01:32.088
to figure out the area.

00:01:32.088 --> 00:01:36.255
This is going to be equal
to pi times three squared.

00:01:37.777 --> 00:01:39.443
Oh and you have to
write parenthesis there.

00:01:39.443 --> 00:01:42.932
Pi, times three, squared,
which is of course

00:01:42.932 --> 00:01:45.285
going to be equal to nine pi.

00:01:45.285 --> 00:01:46.900
So for this particular example,

00:01:46.900 --> 00:01:48.972
when the circumference is six pi units,

00:01:48.972 --> 00:01:50.818
we're able to figure out that the area,

00:01:50.818 --> 00:01:54.034
this is actually going to
be nine pi square units,

00:01:54.034 --> 00:01:56.028
or I could write units squared.

00:01:56.028 --> 00:01:57.812
Cause we're squaring the radius.

00:01:57.812 --> 00:01:59.868
The radius is three units,

00:01:59.868 --> 00:02:03.525
so you square that, you
get the units squared.

00:02:03.525 --> 00:02:06.428
Now let's see if we can come
up with a general formula.

00:02:06.428 --> 00:02:10.595
So we know that circumference
is equal to two pi r.

00:02:12.011 --> 00:02:15.928
And we know that area is
equal to pi r squared.

00:02:19.129 --> 00:02:20.905
Can we come up with an expression

00:02:20.905 --> 00:02:23.923
or a formula that relates directly between

00:02:23.923 --> 00:02:25.578
circumference and area?

00:02:25.578 --> 00:02:28.289
And I'll give you a hint, solve for,

00:02:28.289 --> 00:02:30.785
you could solve for r right over here,

00:02:30.785 --> 00:02:34.559
and substitute back into
this equation, or vice versa.

00:02:34.559 --> 00:02:37.041
Pause the video, see if you can do that.

00:02:37.041 --> 00:02:38.350
Alright, so let's do it over here.

00:02:38.350 --> 00:02:39.457
Let's solve for r.

00:02:39.457 --> 00:02:42.490
If we divide both sides by two pi,

00:02:42.490 --> 00:02:44.077
do it in another color,

00:02:44.077 --> 00:02:46.613
so if we divide both sides by two pi,

00:02:46.613 --> 00:02:48.917
and this is exactly what we did up here,

00:02:48.917 --> 00:02:50.175
what are we left with?

00:02:50.175 --> 00:02:52.580
We're left with on the right hand side,

00:02:52.580 --> 00:02:56.150
r is equal to c the
circumference divided by two pi,

00:02:56.150 --> 00:03:00.317
the radius is equal to the
circumference over two pi.

00:03:01.432 --> 00:03:03.621
And so when we're figuring out the area,

00:03:03.621 --> 00:03:07.788
area, remember, is equal to
pi times our radius squared.

00:03:09.240 --> 00:03:10.899
But we know that our
radius could be written

00:03:10.899 --> 00:03:13.459
as circumference divided by two pi.

00:03:13.459 --> 00:03:16.523
So instead of radius I'll write
circumference over two pi.

00:03:16.523 --> 00:03:18.273
Remember, we want to relate between area

00:03:18.273 --> 00:03:20.083
and circumference.

00:03:20.083 --> 00:03:22.313
And so what is this going to be equal to?

00:03:22.313 --> 00:03:26.480
We get area is equal to pi
times circumference squared,

00:03:29.729 --> 00:03:32.979
over two pi squared is four pi squared.

00:03:35.206 --> 00:03:36.608
Now let's see, we have a pi,

00:03:36.608 --> 00:03:39.988
if we multiplied this out,
we'd have a pi in the numerator

00:03:39.988 --> 00:03:41.401
and a pi in the denominator,

00:03:41.401 --> 00:03:43.838
or two pi's in the
denominator being multiplied.

00:03:43.838 --> 00:03:46.021
So pi divided by pi squared,

00:03:46.021 --> 00:03:50.303
is just one over pi, and
so there you have it.

00:03:50.303 --> 00:03:54.470
Area is equal to circumference
squared divided by four pi.

00:03:57.356 --> 00:03:58.843
Let me write that down.

00:03:58.843 --> 00:04:01.867
So this is a neat, you don't
tend to learn this formula,

00:04:01.867 --> 00:04:03.651
but it's cool that we
were able to derive it.

00:04:03.651 --> 00:04:07.616
Area is equal to circumference
squared over four pi.

00:04:07.616 --> 00:04:09.068
And we can go the other way around.

00:04:09.068 --> 00:04:10.813
Given an area, how do we
figure out circumference?

00:04:10.813 --> 00:04:12.434
Well you could just put
the numbers in here,

00:04:12.434 --> 00:04:14.093
or you could just solve for c.

00:04:14.093 --> 00:04:17.176
Let's multiply both sides by four pi.

00:04:18.437 --> 00:04:20.750
Let's multiply both sides by four pi,

00:04:20.750 --> 00:04:23.207
and if we do that, what do we get?

00:04:23.207 --> 00:04:27.102
We would get four pi
times the area is equal to

00:04:27.102 --> 00:04:29.054
our circumference squared

00:04:29.054 --> 00:04:30.395
and then to solve for the circumference

00:04:30.395 --> 00:04:33.133
we just take the square
root of both sides.

00:04:33.133 --> 00:04:37.947
So you would get the square
root of four pi times the area

00:04:37.947 --> 00:04:40.280
is equal to r circumference.

00:04:41.755 --> 00:04:43.932
And you could simplify this
a little bit if you wanted.

00:04:43.932 --> 00:04:45.713
You could take the four
out of the radical,

00:04:45.713 --> 00:04:46.699
but this is pretty neat,

00:04:46.699 --> 00:04:49.874
how you can relate circumference and area.


Area of an isosceles triangle

WEBVTT
Kind: captions
Language: en

00:00:00.607 --> 00:00:02.229
- [Tutor] Pause this video
and see if you can find

00:00:02.229 --> 00:00:04.633
the area of this triangle,

00:00:04.633 --> 00:00:06.583
and I'll give you two hints.

00:00:06.583 --> 00:00:09.375
Recognize, this is an isosceles triangle,

00:00:09.375 --> 00:00:12.033
and another hint is that
the Pythagorean Theorem

00:00:12.033 --> 00:00:13.366
might be useful.

00:00:14.254 --> 00:00:16.762
Alright, now let's work
through this together.

00:00:16.762 --> 00:00:20.042
So, we might all remember
that the area of a triangle

00:00:20.042 --> 00:00:24.701
is equal to one half times
our base times our height.

00:00:24.701 --> 00:00:25.945
They give us our base.

00:00:25.945 --> 00:00:28.430
Our base right over here is,

00:00:28.430 --> 00:00:29.680
our base is 10.

00:00:31.207 --> 00:00:32.859
But what is our height?

00:00:32.859 --> 00:00:34.200
Our height would be,

00:00:34.200 --> 00:00:35.901
let me do this in another color,

00:00:35.901 --> 00:00:40.011
our height would be the length
of this line right over here.

00:00:40.011 --> 00:00:41.715
So, if we can figure that out,

00:00:41.715 --> 00:00:44.858
then we can calculate what
one half times the base 10

00:00:44.858 --> 00:00:46.498
times the height is.

00:00:46.498 --> 00:00:49.154
But how do we figure out this height?

00:00:49.154 --> 00:00:51.484
Well, this is where
it's useful to recognize

00:00:51.484 --> 00:00:53.995
that this is an isosceles triangle.

00:00:53.995 --> 00:00:57.521
An isosceles triangle has
two sides that are the same.

00:00:57.521 --> 00:01:01.688
And so, these base angles are
also going to be congruent.

00:01:02.529 --> 00:01:06.174
And so, and if we drop an
altitude right over here

00:01:06.174 --> 00:01:08.197
which is the whole
point, that's the height,

00:01:08.197 --> 00:01:12.090
we know that this is, these
are going to be right angles.

00:01:12.090 --> 00:01:14.134
And so, if we have two triangles

00:01:14.134 --> 00:01:15.808
where two of the angles are the same,

00:01:15.808 --> 00:01:18.173
we know that the third angle
is going to be the same.

00:01:18.173 --> 00:01:21.196
So, that is going to be congruent to that.

00:01:21.196 --> 00:01:23.660
And so, if you have two triangles,

00:01:23.660 --> 00:01:26.620
and this might be obvious
already to you intuitively,

00:01:26.620 --> 00:01:29.006
where look, I have two angles in common

00:01:29.006 --> 00:01:31.566
and the side in between them is common,

00:01:31.566 --> 00:01:33.697
it's the same length,

00:01:33.697 --> 00:01:35.730
well that means that these are going to be

00:01:35.730 --> 00:01:37.829
congruent triangles.

00:01:37.829 --> 00:01:39.672
Now, what's useful about
that is if we recognize

00:01:39.672 --> 00:01:41.543
that these are congruent triangles,

00:01:41.543 --> 00:01:43.635
notice that they both have a side 13,

00:01:43.635 --> 00:01:46.397
they both have a side, whatever
this length in blue is.

00:01:46.397 --> 00:01:49.234
And then, they're both
going to have a side length

00:01:49.234 --> 00:01:51.151
that's half of this 10.

00:01:52.571 --> 00:01:55.383
So, this is going to be five,
and this is going to be five.

00:01:55.383 --> 00:01:57.112
How was I able to deduce that?

00:01:57.112 --> 00:01:59.293
You might just say, oh that
feels intuitively right.

00:01:59.293 --> 00:02:00.650
I was a little bit more rigorous here,

00:02:00.650 --> 00:02:03.420
where I said these are
two congruent triangles,

00:02:03.420 --> 00:02:06.143
then we're going to split this 10 in half

00:02:06.143 --> 00:02:07.736
because this is going to be equal to that

00:02:07.736 --> 00:02:09.515
and they add up to 10.

00:02:09.515 --> 00:02:12.283
Alright, now we can use
the Pythagorean Theorem

00:02:12.283 --> 00:02:16.072
to figure out the length of
this blue side or the height.

00:02:16.072 --> 00:02:19.658
If we call this h, the
Pythagorean Theorem tells us

00:02:19.658 --> 00:02:23.221
that h squared plus five
squared is equal to 13 squared.

00:02:23.221 --> 00:02:25.554
H squared plus five squared,

00:02:27.131 --> 00:02:31.589
plus five squared is going
to be equal to 13 squared,

00:02:31.589 --> 00:02:33.191
is going to be equal to our longest side,

00:02:33.191 --> 00:02:35.472
our hypotenuse squared.

00:02:35.472 --> 00:02:36.387
And so, let's see.

00:02:36.387 --> 00:02:37.970
Five squared is 25.

00:02:40.444 --> 00:02:41.944
13 squared is 169.

00:02:44.491 --> 00:02:47.808
We can subtract 25 from both sides

00:02:47.808 --> 00:02:49.949
to isolate the h squared.

00:02:49.949 --> 00:02:51.599
So, let's do that.

00:02:51.599 --> 00:02:53.728
And what are we left with?

00:02:53.728 --> 00:02:57.420
We are left with h squared is equal to

00:02:57.420 --> 00:03:00.753
these canceled out, 169 minus 25 is 144.

00:03:03.483 --> 00:03:04.843
Now, if you're doing it
purely mathematically,

00:03:04.843 --> 00:03:07.017
you say, oh h could be plus or minus 12,

00:03:07.017 --> 00:03:08.322
but we're dealing with the distance,

00:03:08.322 --> 00:03:10.598
so we'll focus on the positive.

00:03:10.598 --> 00:03:15.298
So, h is going to be equal
to the principal root of 144.

00:03:15.298 --> 00:03:17.084
So, h is equal to 12.

00:03:17.084 --> 00:03:18.043
Now, we aren't done.

00:03:18.043 --> 00:03:19.149
Remember, they don't want us to just

00:03:19.149 --> 00:03:20.307
figure out the height here,

00:03:20.307 --> 00:03:22.472
they want us to figure out the area.

00:03:22.472 --> 00:03:25.389
Area is one half base times height.

00:03:26.289 --> 00:03:27.430
Well, we already figured out

00:03:27.430 --> 00:03:31.052
that our base is this 10 right over here,

00:03:31.052 --> 00:03:32.809
let me do this in another color.

00:03:32.809 --> 00:03:36.309
So, our base is that distance which is 10,

00:03:37.443 --> 00:03:39.533
and now we know our height.

00:03:39.533 --> 00:03:40.950
Our height is 12.

00:03:42.328 --> 00:03:45.925
So, now we just have to compute
one half times 10 times 12.

00:03:45.925 --> 00:03:47.875
Well, that's just going to be equal to

00:03:47.875 --> 00:03:50.096
one half times 10 is five,

00:03:50.096 --> 00:03:52.072
times 12 is 60,

00:03:52.072 --> 00:03:55.947
60 square units, whatever
our units happen to be.

00:03:55.947 --> 00:03:57.364
That is our area.


Triangle missing side example

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:01.860
- [Instructor] The triangle shown below

00:00:01.860 --> 00:00:05.400
has an area of 75 square units.

00:00:05.400 --> 00:00:07.560
Find the missing side.

00:00:07.560 --> 00:00:10.770
So pause the video and see
if you can find the length

00:00:10.770 --> 00:00:13.083
of this missing side.

00:00:14.040 --> 00:00:15.780
Alright, now let's work
through this together.

00:00:15.780 --> 00:00:20.130
They give us the area,
they give us this side,

00:00:20.130 --> 00:00:24.810
right over here, this 11,
they give us this length 10,

00:00:24.810 --> 00:00:26.730
which if we rotate this triangle,

00:00:26.730 --> 00:00:28.500
you could view it as an altitude.

00:00:28.500 --> 00:00:30.150
And in fact, let me do that.

00:00:30.150 --> 00:00:31.470
Let me rotate this triangle

00:00:31.470 --> 00:00:33.450
because then I think it
might jump out at you

00:00:33.450 --> 00:00:35.340
how we can tackle this.

00:00:35.340 --> 00:00:39.000
So let me copy and let me paste it.

00:00:39.000 --> 00:00:42.180
So if I move it here,
but I'm gonna rotate it.

00:00:42.180 --> 00:00:45.674
So if I rotate the, oh,

00:00:45.674 --> 00:00:48.390
that is our rotated triangle

00:00:48.390 --> 00:00:50.370
and now it might be a little bit clearer

00:00:50.370 --> 00:00:51.480
what we're talking about.

00:00:51.480 --> 00:00:53.700
This length x that we want to figure out.

00:00:53.700 --> 00:00:56.340
This is our base and
they give us our height

00:00:56.340 --> 00:00:57.420
and they give us our area.

00:00:57.420 --> 00:01:01.020
And we know how base, height
and area relate for a triangle.

00:01:01.020 --> 00:01:04.765
We know that area is equal to 1/2

00:01:04.765 --> 00:01:09.150
times the base times the height.

00:01:09.150 --> 00:01:11.400
And they tell us, they tell us

00:01:11.400 --> 00:01:13.710
that our area is 75 unit squared.

00:01:13.710 --> 00:01:18.710
So this is 75 is equal to 1/2.

00:01:19.050 --> 00:01:23.580
What is our base? Our
base is the variable x.

00:01:23.580 --> 00:01:26.490
So let's just write
that down. 1/2 times x.

00:01:26.490 --> 00:01:28.140
And then what is our height?

00:01:28.140 --> 00:01:30.120
Well, our height is actually the 10.

00:01:30.120 --> 00:01:33.900
If x is the length of
our base, then the height

00:01:33.900 --> 00:01:35.130
of our triangle is gonna be 10.

00:01:35.130 --> 00:01:37.260
We actually don't even
need to use this 11.

00:01:37.260 --> 00:01:39.268
They're putting that there
just to distract you.

00:01:39.268 --> 00:01:44.268
So this is going to be
our height, times 10.

00:01:44.580 --> 00:01:47.940
So 75 is equal to 1/2 times x times 10.

00:01:47.940 --> 00:01:49.530
Or lemme just rewrite it this way.

00:01:49.530 --> 00:01:54.530
We could say 75 is equal to 1/2 times 10

00:01:54.540 --> 00:01:59.540
is equal to 5 times x,
is equal to 5 times,

00:01:59.740 --> 00:02:04.600
lemme do the x in that same
color is equal to 5 times x.

00:02:06.240 --> 00:02:08.220
So what is x going to be?

00:02:08.220 --> 00:02:09.690
There's a couple of ways
you could think about it.

00:02:09.690 --> 00:02:13.020
You could say 5 times what is equal to 75

00:02:13.020 --> 00:02:14.910
and you might be able to figure that out.

00:02:14.910 --> 00:02:18.060
You might say, okay, let's
see, 5 times 10 is 50.

00:02:18.060 --> 00:02:19.560
And then let's see, I need another 25.

00:02:19.560 --> 00:02:22.890
So I'd put another 5 there.
So it's really 5 times 15.

00:02:22.890 --> 00:02:25.620
Or you could do it a little
bit more systematically.

00:02:25.620 --> 00:02:29.430
You can divide both sides by
what you're multiplying by x.

00:02:29.430 --> 00:02:32.970
So if you divide this side
by 5, 5 times x divided

00:02:32.970 --> 00:02:35.700
by 5, well you're just
going to have an x leftover.

00:02:35.700 --> 00:02:37.650
But you can't, these
two things were equal.

00:02:37.650 --> 00:02:39.030
So you can't just do it to one side,

00:02:39.030 --> 00:02:40.410
you have to do it to both sides.

00:02:40.410 --> 00:02:42.810
So you have to divide both sides by 5.

00:02:42.810 --> 00:02:47.460
And what's 75 divided
by 5? Well that is 15.

00:02:47.460 --> 00:02:52.460
So you get x is equal
to, x is equal to 15.

00:02:53.880 --> 00:02:56.760
And you can verify that,
if x is equal to 15,

00:02:56.760 --> 00:03:01.260
base times height, times
1/2, well it's 15 times 10

00:03:01.260 --> 00:03:03.570
times 1/2 or 15 times 5,

00:03:03.570 --> 00:03:06.633
which is going to be 75 square units.


Recognizing common 3D shapes

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.260
- [Narrator] So I have five

00:00:01.260 --> 00:00:03.900
three dimensional shapes over here,

00:00:03.900 --> 00:00:06.930
and I also have five names for them.

00:00:06.930 --> 00:00:09.000
And what I want you to
do is pause this video

00:00:09.000 --> 00:00:11.610
and think about which of these
shapes is a square pyramid,

00:00:11.610 --> 00:00:13.290
which of these is a rectangular prism?

00:00:13.290 --> 00:00:15.150
Which one is a triangular prism?

00:00:15.150 --> 00:00:17.943
Which one is a sphere and
which one is a cylinder?

00:00:18.990 --> 00:00:20.790
All right, now let's just
work through this together.

00:00:20.790 --> 00:00:22.860
And really this is just something to know

00:00:22.860 --> 00:00:25.860
because some people in various subjects

00:00:25.860 --> 00:00:28.500
will sometimes throw out these words.

00:00:28.500 --> 00:00:29.970
So first let's start, let's think about

00:00:29.970 --> 00:00:32.580
which one is the square pyramid.

00:00:32.580 --> 00:00:34.620
So this shape here on the right

00:00:34.620 --> 00:00:36.360
clearly looks like a pyramid.

00:00:36.360 --> 00:00:38.910
And the thing that makes
it a square pyramid

00:00:38.910 --> 00:00:42.780
is the idea that its base is a square.

00:00:42.780 --> 00:00:45.150
Let me see if I can shade this in.

00:00:45.150 --> 00:00:49.607
So the base of this right
over here is a square.

00:00:50.550 --> 00:00:54.750
So that's where the
square pyramid comes from.

00:00:54.750 --> 00:00:56.280
If it's a triangular pyramid,

00:00:56.280 --> 00:00:58.380
then the base would only have three sides,

00:00:58.380 --> 00:01:00.540
but this is a square pyramid.

00:01:00.540 --> 00:01:05.540
So let me move this over there.

00:01:05.730 --> 00:01:09.000
So that is a square pyramid.

00:01:09.000 --> 00:01:11.070
All right, let's do the next one.

00:01:11.070 --> 00:01:14.070
Which one is a rectangular prism?

00:01:14.070 --> 00:01:18.300
A prism you can view as a
three dimensional object.

00:01:18.300 --> 00:01:22.140
And since it's saying rectangular prism,

00:01:22.140 --> 00:01:24.960
well the best one here
that is rectangular,

00:01:24.960 --> 00:01:26.190
notice this shape right over here

00:01:26.190 --> 00:01:28.380
is rectangular on every side.

00:01:28.380 --> 00:01:29.430
And it's three dimensions.

00:01:29.430 --> 00:01:30.600
It is a prism.

00:01:30.600 --> 00:01:33.480
And so this is the rectangular prism.

00:01:33.480 --> 00:01:35.640
So let's move that over.

00:01:35.640 --> 00:01:40.583
So that is our rectangular prism.

00:01:42.390 --> 00:01:47.390
Now, triangular prism, so
this one is interesting.

00:01:47.610 --> 00:01:49.770
If you look at this shape right over here,

00:01:49.770 --> 00:01:52.260
it has sides that are triangles.

00:01:52.260 --> 00:01:55.800
You see this up here is a triangle.

00:01:55.800 --> 00:01:58.710
And you see this down here is a triangle,

00:01:58.710 --> 00:02:01.350
although they're connected
by these rectangular sides.

00:02:01.350 --> 00:02:05.250
And this is actually what's
called a triangular prism.

00:02:05.250 --> 00:02:07.650
In a square pyramid, the faces

00:02:07.650 --> 00:02:09.930
which are triangles all meet at a point.

00:02:09.930 --> 00:02:11.370
In a triangular prism,

00:02:11.370 --> 00:02:14.760
you have two opposite facing
sides that are triangles.

00:02:14.760 --> 00:02:16.980
And you'll see all of these
in different orientations,

00:02:16.980 --> 00:02:18.240
in different, you know,

00:02:18.240 --> 00:02:20.280
sometimes it'll be resting like this,

00:02:20.280 --> 00:02:21.870
sometimes it'll be on its side.

00:02:21.870 --> 00:02:24.030
But this is a triangular prism.

00:02:24.030 --> 00:02:25.680
Let me-

00:02:25.680 --> 00:02:30.633
so that is a triangular prism.

00:02:32.700 --> 00:02:34.410
And so we have two left.

00:02:34.410 --> 00:02:36.210
You might be able to figure these out.

00:02:36.210 --> 00:02:38.280
Which one is a sphere?

00:02:38.280 --> 00:02:41.520
Well, when I think sphere,
I think of like a globe.

00:02:41.520 --> 00:02:44.673
And so this choice right over
here is clearly a sphere.

00:02:46.020 --> 00:02:47.823
So let me move that over.

00:02:49.920 --> 00:02:51.210
Let me move it over.

00:02:51.210 --> 00:02:54.630
Okay, so that one is a sphere.

00:02:54.630 --> 00:02:55.860
And then last but not least,

00:02:55.860 --> 00:02:59.403
a thing that looks like
a can is a cylinder.

00:03:00.750 --> 00:03:03.030
So let me move that over.

00:03:03.030 --> 00:03:07.710
So that is a cylinder and we're all done.

00:03:07.710 --> 00:03:10.140
And so these are just useful
names to know for shapes.

00:03:10.140 --> 00:03:13.380
Cylinder is, you're connecting,

00:03:13.380 --> 00:03:15.540
you have these two faces that are circular

00:03:15.540 --> 00:03:17.400
and then they're opposite each other.

00:03:17.400 --> 00:03:19.020
It's kind of like a triangular prism.

00:03:19.020 --> 00:03:20.670
But instead of having these hard edges,

00:03:20.670 --> 00:03:22.020
'cause you have a triangle,

00:03:22.020 --> 00:03:24.390
you have a base that is a circle

00:03:24.390 --> 00:03:26.433
and a top that is a circle.


Example finding distance with Pythagorean theorem

WEBVTT
Kind: captions
Language: en

00:00:00.271 --> 00:00:02.469
- We are asked what is the distance

00:00:02.469 --> 00:00:05.160
between the following points.

00:00:05.160 --> 00:00:08.381
Pause this video and see
if you can figure it out.

00:00:08.381 --> 00:00:10.433
There's multiple ways to think about it.

00:00:10.433 --> 00:00:12.925
The way I think about it
is really to try to draw

00:00:12.925 --> 00:00:15.800
a right triangle where these points,

00:00:15.800 --> 00:00:18.881
where the line that connects
these points is the hypotenuse

00:00:18.881 --> 00:00:21.462
and then we can just use
the Pythagorean Theorem.

00:00:21.462 --> 00:00:24.220
Let me show you what I am talking about.

00:00:24.220 --> 00:00:27.137
Let me draw a right triangle, here.

00:00:28.249 --> 00:00:31.264
That is the height of my right triangle

00:00:31.264 --> 00:00:34.847
and this is the width
of my right triangle.

00:00:35.720 --> 00:00:37.772
Then the hypotenuse will
connect these two points.

00:00:37.772 --> 00:00:40.441
I could use my little
ruler tool here to connect

00:00:40.441 --> 00:00:44.024
that point and that
point right over there.

00:00:45.029 --> 00:00:47.191
I'll color it in orange.

00:00:47.191 --> 00:00:49.309
There you have it.

00:00:49.309 --> 00:00:50.478
There you have it.

00:00:50.478 --> 00:00:53.302
I have a right triangle
where the line that connects

00:00:53.302 --> 00:00:57.361
those two points is the
hypotenuse of that right triangle.

00:00:57.361 --> 00:00:58.861
Why is that useful?

00:00:58.861 --> 00:01:00.802
From this, can you pause
the video and figure out

00:01:00.802 --> 00:01:03.537
the length of that orange
line, which is the distance

00:01:03.537 --> 00:01:06.273
between those two points?

00:01:06.273 --> 00:01:09.405
What is the length of this red line?

00:01:09.405 --> 00:01:11.567
You could see it on this grid, here.

00:01:11.567 --> 00:01:13.354
This is equal to two.

00:01:13.354 --> 00:01:15.957
It's exactly two spaces, and
you could even think about it

00:01:15.957 --> 00:01:17.391
in terms of coordinates.

00:01:17.391 --> 00:01:19.619
The coordinate of this point up here

00:01:19.619 --> 00:01:22.036
is negative five comma eight.

00:01:23.016 --> 00:01:25.751
Negative five comma eight.

00:01:25.751 --> 00:01:29.334
The coordinate here is
X is four, Y is six.

00:01:30.339 --> 00:01:33.825
Four comma six, and so
the coordinate over here

00:01:33.825 --> 00:01:37.992
is going to have the same
Y coordinate as this point.

00:01:39.494 --> 00:01:41.413
This is going to be comma six.

00:01:41.413 --> 00:01:43.134
It's going to have the same
X coordinate as this point.

00:01:43.134 --> 00:01:45.560
This is going to be
negative five comma six.

00:01:45.560 --> 00:01:48.362
Notice, you're only
changing in the Y direction

00:01:48.362 --> 00:01:50.700
and you're changing by two.

00:01:50.700 --> 00:01:53.612
What's the length of this line?

00:01:53.612 --> 00:01:55.884
You could count it out, one, two, three,

00:01:55.884 --> 00:01:59.171
four, five, six, seven, eight, nine.

00:01:59.171 --> 00:02:01.752
It's nine, or you could even say hey look,

00:02:01.752 --> 00:02:04.576
we're only changing in the X value.

00:02:04.576 --> 00:02:06.186
We're going from negative five,

00:02:06.186 --> 00:02:09.076
X equals negative five, to X equals four.

00:02:09.076 --> 00:02:11.480
We're going to increase by nine.

00:02:11.480 --> 00:02:13.620
All of that just sets us up so that

00:02:13.620 --> 00:02:15.605
we can use the Pythagorean Theorem.

00:02:15.605 --> 00:02:19.772
If we call this C, we know
that A squared plus B squared

00:02:21.936 --> 00:02:25.290
is equal to C squared, or we
could say that two squared ...

00:02:25.290 --> 00:02:26.834
Let me do it over here.

00:02:26.834 --> 00:02:28.996
Use that same red color.

00:02:28.996 --> 00:02:33.079
Two squared plus nine
squared, plus nine squared,

00:02:34.885 --> 00:02:37.930
is going to be equal to
our hypotenuse square,

00:02:37.930 --> 00:02:40.488
which I'm just calling C, is
going to be equal to C squared,

00:02:40.488 --> 00:02:41.636
which is really the distance.

00:02:41.636 --> 00:02:43.400
That's what we're trying to figure out.

00:02:43.400 --> 00:02:47.567
Two squared, that is four,
plus nine squared is 81.

00:02:50.173 --> 00:02:53.901
That's going to be equal to C squared.

00:02:53.901 --> 00:02:56.568
We get C squared is equal to 85.

00:02:57.430 --> 00:02:59.847
C squared is equal to 85 or C

00:03:00.739 --> 00:03:04.423
is equal to the principal root of 85.

00:03:04.423 --> 00:03:06.916
Can I simplify that a little bit?

00:03:06.916 --> 00:03:07.930
Let's see.

00:03:07.930 --> 00:03:11.063
How many times does five go into 85?

00:03:11.063 --> 00:03:14.146
It goes, let's see, it goes 17 times.

00:03:16.092 --> 00:03:17.857
Neither of those are perfect squares.

00:03:17.857 --> 00:03:19.997
Yeah, that's 50 plus 35.

00:03:19.997 --> 00:03:22.284
Yeah, I think that's about
as simple as I can write it.

00:03:22.284 --> 00:03:23.960
If you wanted to express it as a decimal,

00:03:23.960 --> 00:03:26.321
you could approximate it by
putting this into a calculator

00:03:26.321 --> 00:03:29.100
and however precise you want
your approximation to be.

00:03:29.100 --> 00:03:31.814
That over here, that's
the length of this line,

00:03:31.814 --> 00:03:33.358
our hypotenuse and our right triangle,

00:03:33.358 --> 00:03:35.056
but more importantly for
the question they're asking,

00:03:35.056 --> 00:03:37.889
the distance between those points.


Partial circle area and arc length

WEBVTT
Kind: captions
Language: en

00:00:00.409 --> 00:00:03.064
- [Instructor] Find the
area of the semicircle.

00:00:03.064 --> 00:00:05.700
So pause this video and see
if you can figure it out.

00:00:05.700 --> 00:00:07.270
So let's see.

00:00:07.270 --> 00:00:09.510
We know that the area of a circle

00:00:09.510 --> 00:00:12.843
is equal to pi times our radius squared.

00:00:14.725 --> 00:00:16.845
So, if we think about the entire circle,

00:00:16.845 --> 00:00:19.167
what is the area going to be?

00:00:19.167 --> 00:00:21.009
Well, they tell us what our radius is.

00:00:21.009 --> 00:00:22.682
Our radius is equal to two,

00:00:22.682 --> 00:00:25.876
so the area, if we're talking
about the whole circle,

00:00:25.876 --> 00:00:30.073
it would be equal to pi times two squared.

00:00:30.073 --> 00:00:31.823
Pi times two squared.

00:00:32.859 --> 00:00:34.737
Two squared is of course two times two,

00:00:34.737 --> 00:00:35.930
which is equal to four,

00:00:35.930 --> 00:00:40.923
so our area is going to
be equal to four times pi.

00:00:40.923 --> 00:00:42.961
Now, I wouldn't put four pi here,

00:00:42.961 --> 00:00:45.251
because that would be the entire circle.

00:00:45.251 --> 00:00:47.885
They want the area of just the semicircle,

00:00:47.885 --> 00:00:50.899
of just this region right over here.

00:00:50.899 --> 00:00:53.163
Well, the semicircle
is half of the circle,

00:00:53.163 --> 00:00:55.215
so if I want the area of the semicircle,

00:00:55.215 --> 00:00:56.651
this is gonna be half this.

00:00:56.651 --> 00:00:58.099
So instead of four pi,

00:00:58.099 --> 00:01:01.999
it is going to be two pi square units.

00:01:01.999 --> 00:01:03.938
That's the area of the semicircle.

00:01:03.938 --> 00:01:06.021
Let's do another example.

00:01:08.110 --> 00:01:09.416
So here, instead of area,

00:01:09.416 --> 00:01:12.839
we're asked to find the arc
length of the partial circle,

00:01:12.839 --> 00:01:15.467
and that's we have here
in this bluish color

00:01:15.467 --> 00:01:18.200
right over here, find this arc length.

00:01:18.200 --> 00:01:20.460
And you can see this
is going three fourths

00:01:20.460 --> 00:01:22.915
of the way around the circle,

00:01:22.915 --> 00:01:25.173
so this arc length is going to be

00:01:25.173 --> 00:01:27.597
three fourths of the circumference.

00:01:27.597 --> 00:01:29.374
So what is the circumference?

00:01:29.374 --> 00:01:31.132
Well, we know the circumference

00:01:31.132 --> 00:01:34.132
is equal to two pi times the radius.

00:01:35.410 --> 00:01:37.950
They tell us what the radius is.

00:01:37.950 --> 00:01:40.552
It's equal to four, so our circumference

00:01:40.552 --> 00:01:43.052
is equal to two pi times four.

00:01:47.734 --> 00:01:50.049
Let's see, we can just change
the order in which we multiply

00:01:50.049 --> 00:01:52.205
so it's two times four times pi.

00:01:52.205 --> 00:01:55.107
This is going to be equal to eight pi.

00:01:55.107 --> 00:01:57.796
This is going to be equal to eight pi.

00:01:57.796 --> 00:02:01.137
Now, that is the circumference
of the entire circle.

00:02:01.137 --> 00:02:03.332
If we care about this arc length,

00:02:03.332 --> 00:02:05.426
it's going to be three fourths times

00:02:05.426 --> 00:02:07.603
the circumference of the entire circle.

00:02:07.603 --> 00:02:10.436
So three over four times eight pi.

00:02:12.298 --> 00:02:14.264
What is that going to be?

00:02:14.264 --> 00:02:17.470
Well, what's three fourths times eight?

00:02:17.470 --> 00:02:22.135
Well, three times eight is
24 divided by four is six.

00:02:22.135 --> 00:02:25.417
So this is going to be equal to six pi.

00:02:25.417 --> 00:02:26.735
Another way to think about it,

00:02:26.735 --> 00:02:28.116
one fourth of eight is two,

00:02:28.116 --> 00:02:31.303
so three fourths is going to be six.

00:02:31.303 --> 00:02:32.462
Or another way to think about it is,

00:02:32.462 --> 00:02:34.941
one fourth of eight pi is two pi,

00:02:34.941 --> 00:02:38.799
and so three of those is
going to be equal to six pi.

00:02:38.799 --> 00:02:43.065
So the arc length of the
partial circle is six pi,

00:02:43.065 --> 00:02:44.062
and once again we knew that

00:02:44.062 --> 00:02:45.695
because it was three
fourths of the way around.

00:02:45.695 --> 00:02:46.945
The way that I knew it was three fourths

00:02:46.945 --> 00:02:48.673
is that this is a 90 degree angle.

00:02:48.673 --> 00:02:51.323
This is 90 degrees, which is one fourth

00:02:51.323 --> 00:02:52.997
of the way around a circle,

00:02:52.997 --> 00:02:54.179
so the arc length that we care about

00:02:54.179 --> 00:02:57.679
is the three fourths of our circumference.


Pythagorean theorem with right triangle

WEBVTT
Kind: captions
Language: en

00:00:00.542 --> 00:00:02.660
- [Instructor] We're asked
to find the value of x

00:00:02.660 --> 00:00:05.112
in the isosceles triangle shown below.

00:00:05.112 --> 00:00:07.543
So that is the base of this triangle.

00:00:07.543 --> 00:00:12.454
So pause this video and see
if you can figure that out.

00:00:12.454 --> 00:00:14.530
Well the key realization to solve this

00:00:14.530 --> 00:00:17.755
is to realize that this
altitude that they dropped,

00:00:17.755 --> 00:00:20.318
this is going to form a right angle here

00:00:20.318 --> 00:00:22.150
and a right angle here

00:00:22.150 --> 00:00:23.216
and notice,

00:00:23.216 --> 00:00:24.533
both of these triangles,

00:00:24.533 --> 00:00:27.222
because this whole thing
is an isosceles triangle,

00:00:27.222 --> 00:00:30.537
we're going to have two
angles that are the same.

00:00:30.537 --> 00:00:32.620
This angle, is the same as that angle.

00:00:32.620 --> 00:00:35.058
Because it's an isosceles triangle,

00:00:35.058 --> 00:00:37.448
this 90 degrees is the
same as that 90 degrees.

00:00:37.448 --> 00:00:39.238
And so the third angle
needs to be the same.

00:00:39.238 --> 00:00:42.387
So that is going to be the same as that

00:00:42.387 --> 00:00:43.613
right over there.

00:00:43.613 --> 00:00:46.622
And since you have two
angles that are the same

00:00:46.622 --> 00:00:49.060
and you have a side between them

00:00:49.060 --> 00:00:49.893
that is the same

00:00:49.893 --> 00:00:53.358
this altitude of 12 is on both triangles,

00:00:53.358 --> 00:00:57.008
we know that both of these
triangles are congruent.

00:00:57.008 --> 00:00:58.903
So they're both going to have 13

00:00:58.903 --> 00:01:00.360
they're going to have one side that's 13,

00:01:00.360 --> 00:01:02.387
one side that is 12

00:01:02.387 --> 00:01:03.787
and so this

00:01:03.787 --> 00:01:06.149
and this side are going to be the same.

00:01:06.149 --> 00:01:09.116
So this is going to be x over two

00:01:09.116 --> 00:01:11.896
and this is going to be x over two.

00:01:11.896 --> 00:01:14.027
And so now we can use that information

00:01:14.027 --> 00:01:15.915
and the fact and the Pythagorean Theorem

00:01:15.915 --> 00:01:17.165
to solve for x.

00:01:18.270 --> 00:01:20.549
Let's use the Pythagorean Theorem

00:01:20.549 --> 00:01:22.847
on this right triangle

00:01:22.847 --> 00:01:24.764
on the right hand side.

00:01:25.822 --> 00:01:27.072
We can say that

00:01:28.218 --> 00:01:29.718
x over two squared

00:01:30.872 --> 00:01:32.495
that's the base right over here

00:01:32.495 --> 00:01:34.195
this side right over here.

00:01:34.195 --> 00:01:36.786
We can write that x over two

00:01:36.786 --> 00:01:37.619
squared

00:01:38.563 --> 00:01:40.457
plus the other side

00:01:40.457 --> 00:01:41.707
plus 12 squared

00:01:42.889 --> 00:01:46.427
is going to be equal to
our hypotenuse squared.

00:01:46.427 --> 00:01:49.228
Is going to be equal to 13 squared.

00:01:49.228 --> 00:01:51.659
This is just the Pythagorean Theorem now.

00:01:51.659 --> 00:01:53.317
And so we can simplify.

00:01:53.317 --> 00:01:55.274
This is going to be x.

00:01:55.274 --> 00:01:56.981
We'll give that the same color.

00:01:56.981 --> 00:02:00.993
This is going to be x squared over four.

00:02:00.993 --> 00:02:03.649
That's just x squared over two squared

00:02:03.649 --> 00:02:04.482
plus 144

00:02:06.533 --> 00:02:09.366
144 is equal to 13 squared is 169.

00:02:12.246 --> 00:02:14.656
Now I can subtract 144 from both sides.

00:02:14.656 --> 00:02:15.868
I'm gonna try to solve for x.

00:02:15.868 --> 00:02:17.679
That's the whole goal here.

00:02:17.679 --> 00:02:21.336
So subtracting 144 from both sides

00:02:21.336 --> 00:02:22.924
and what do we get?

00:02:22.924 --> 00:02:25.342
On the left hand side, we have x squared

00:02:25.342 --> 00:02:28.342
over four is equal to 169 minus 144.

00:02:30.071 --> 00:02:33.311
Let's see, 69 minus 44 is 25.

00:02:33.311 --> 00:02:36.647
So this is going to be equal to 25.

00:02:36.647 --> 00:02:39.810
We can multiply both sides by four

00:02:39.810 --> 00:02:41.948
to isolate the x squared.

00:02:41.948 --> 00:02:45.543
And so we get x squared
is equal to 25 times four

00:02:45.543 --> 00:02:47.096
is equal to 100.

00:02:47.096 --> 00:02:49.110
Now, if you're just looking
this purely mathematically

00:02:49.110 --> 00:02:52.209
and say, x could be
positive or negative 10.

00:02:52.209 --> 00:02:53.756
But since we're dealing with distances,

00:02:53.756 --> 00:02:56.089
we know that we want the
positive value of it.

00:02:56.089 --> 00:02:59.217
So x is equal to the principle root of 100

00:02:59.217 --> 00:03:01.627
which is equal to positive 10.

00:03:01.627 --> 00:03:02.460
So there you have it.

00:03:02.460 --> 00:03:04.316
We have solved for x.

00:03:04.316 --> 00:03:06.650
This distance right here, the whole thing,

00:03:06.650 --> 00:03:09.206
the whole thing is
going to be equal to 10.

00:03:09.206 --> 00:03:11.275
Half of that is going to be five.

00:03:11.275 --> 00:03:14.271
So if we just looked at
this length over here.

00:03:14.271 --> 00:03:16.576
I'm doing that in the
same column, let me see.

00:03:16.576 --> 00:03:19.307
So this length right over here,

00:03:19.307 --> 00:03:20.331
that's going to be five

00:03:20.331 --> 00:03:22.309
and indeed, five squared plus 12 squared,

00:03:22.309 --> 00:03:24.420
that's 25 plus 144

00:03:24.420 --> 00:03:26.003
is 169, 13 squared.

00:03:27.025 --> 00:03:29.206
So the key of realization here is

00:03:29.206 --> 00:03:30.432
isosceles triangle,

00:03:30.432 --> 00:03:34.159
the altitudes splits it into
two congruent right triangles

00:03:34.159 --> 00:03:37.384
and so it also splits this base into two.

00:03:37.384 --> 00:03:39.042
So this is x over two

00:03:39.042 --> 00:03:40.114
and this is x over two.

00:03:40.114 --> 00:03:41.396
And we use that information

00:03:41.396 --> 00:03:43.180
and the Pythagorean Theorem

00:03:43.180 --> 00:03:44.430
to solve for x.


How volume changes from changing dimensions

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:02.130
- [Narrator] I have a
rectangular prism here

00:00:02.130 --> 00:00:04.290
and we're given two of the dimensions.

00:00:04.290 --> 00:00:06.720
The width is 2, the depth is 3.

00:00:06.720 --> 00:00:09.810
And this height here, we're
just representing with an H.

00:00:09.810 --> 00:00:11.670
And what we're gonna do in
this video is think about

00:00:11.670 --> 00:00:15.120
how does the volume of this
rectangular prism change

00:00:15.120 --> 00:00:17.340
as we change the height?

00:00:17.340 --> 00:00:19.650
So let's make a little table here.

00:00:19.650 --> 00:00:23.880
So let me make my table.

00:00:23.880 --> 00:00:25.440
So this is going to be our height

00:00:25.440 --> 00:00:28.680
and this is going to be
our volume, V for volume.

00:00:28.680 --> 00:00:31.320
And so let's say that the height is 5.

00:00:31.320 --> 00:00:33.450
What is the volume going to be?

00:00:33.450 --> 00:00:36.630
Pause this video and see
if you can figure it out.

00:00:36.630 --> 00:00:40.230
Well, the volume is just going
to be the base times height,

00:00:40.230 --> 00:00:42.810
times depth, or you
could say it's gonna be

00:00:42.810 --> 00:00:44.310
the area of the square.

00:00:44.310 --> 00:00:46.260
So it's the width times the depth,

00:00:46.260 --> 00:00:49.830
which is 6 times the height.

00:00:49.830 --> 00:00:53.520
So that would be 2 times 3 times 5,

00:00:53.520 --> 00:00:58.520
so 2 times 3 times 5, which
is equal to 6 times 5,

00:00:59.400 --> 00:01:03.750
which is equal to 30, 30 cubic units.

00:01:03.750 --> 00:01:05.790
We're assuming that these
are given in some units,

00:01:05.790 --> 00:01:08.100
so this would be the units cubed.

00:01:08.100 --> 00:01:10.380
All right, now let's think about it

00:01:10.380 --> 00:01:12.870
if we were to double the height,

00:01:12.870 --> 00:01:16.080
what is going to happen to our volume?

00:01:16.080 --> 00:01:17.610
So if we double the height,

00:01:17.610 --> 00:01:19.710
our height is 10, what is the volume?

00:01:19.710 --> 00:01:22.740
Pause this video and see
if you can figure it out.

00:01:22.740 --> 00:01:26.730
Well, in this situation we're
still gonna have 2 times 3,

00:01:26.730 --> 00:01:31.730
2 times 3 times our new height, times 10.

00:01:32.940 --> 00:01:36.630
So now, it's going to be 6
times 10, which is equal to 60.

00:01:36.630 --> 00:01:39.750
Notice when we doubled the height,

00:01:39.750 --> 00:01:42.240
if we just double one dimension,

00:01:42.240 --> 00:01:45.510
we are going to double the volume.

00:01:45.510 --> 00:01:47.070
Let's see if that holds up.

00:01:47.070 --> 00:01:48.900
Let's double it again.

00:01:48.900 --> 00:01:52.800
So what happens when
our height is 20 units?

00:01:52.800 --> 00:01:55.890
Well, here our volume is
still gonna be 2 times 3

00:01:55.890 --> 00:02:00.890
times 20, 2 times 3 times 20,

00:02:02.190 --> 00:02:07.190
which is equal to 6 times
20, which is equal to 120.

00:02:07.500 --> 00:02:10.290
So once again, if you double
one of the dimensions,

00:02:10.290 --> 00:02:14.460
in this case, the height,
it doubles the volume.

00:02:14.460 --> 00:02:16.050
And you could think of it the other way.

00:02:16.050 --> 00:02:18.930
If you were to halve, if you
were to go from 20 to 10.

00:02:18.930 --> 00:02:22.890
So if you halve one of the
dimensions, it halves the volume,

00:02:22.890 --> 00:02:25.413
you go from 120 to 60.

00:02:26.340 --> 00:02:28.140
Now, let's think about
something interesting.

00:02:28.140 --> 00:02:29.880
Let's think about what happens

00:02:29.880 --> 00:02:33.810
if we double two of the dimensions.

00:02:33.810 --> 00:02:38.810
So let's say, so we know, I'll
just draw these really fast.

00:02:39.450 --> 00:02:41.860
We know that if we have a situation

00:02:42.750 --> 00:02:47.750
where we have 2 by 3 and this height is 5,

00:02:49.770 --> 00:02:54.360
we know the volume here
is 30, 30 cubic units.

00:02:54.360 --> 00:02:56.160
But now let's double
two of the dimensions.

00:02:56.160 --> 00:02:59.460
Let's make this into a 10
and let's make this into a 4.

00:02:59.460 --> 00:03:01.830
So it's gonna look like this,

00:03:01.830 --> 00:03:03.780
and then this is gonna be a 4.

00:03:03.780 --> 00:03:06.033
This is still going to be a 3,

00:03:06.960 --> 00:03:09.390
and our height is going to be a 10.

00:03:09.390 --> 00:03:12.150
So it's gonna look something like this.

00:03:12.150 --> 00:03:14.640
So our height is going to be a 10.

00:03:14.640 --> 00:03:17.070
I haven't drawn it perfectly to scale,

00:03:17.070 --> 00:03:19.380
but hopefully, you get the idea.

00:03:19.380 --> 00:03:23.010
So this is our height at 10.

00:03:23.010 --> 00:03:24.630
What is the volume gonna be now?

00:03:24.630 --> 00:03:27.300
Pause this video and see
if you can figure it out.

00:03:27.300 --> 00:03:32.300
Well, 4 times 3 is 12 times 10 is 120.

00:03:35.430 --> 00:03:39.420
So notice when we doubled
two of the dimensions,

00:03:39.420 --> 00:03:41.820
we actually quadrupled.

00:03:41.820 --> 00:03:44.490
We actually quadrupled our total volume.

00:03:44.490 --> 00:03:45.450
Think about, pause this video

00:03:45.450 --> 00:03:47.283
and think about why did that happen?

00:03:48.510 --> 00:03:51.690
Well, if you double one
dimension, you double the volume,

00:03:51.690 --> 00:03:53.700
but here we're doubling one dimension

00:03:53.700 --> 00:03:54.780
and then another dimension.

00:03:54.780 --> 00:03:58.470
So you are multiplying by 2 twice.

00:03:58.470 --> 00:03:59.880
So think about what would happen

00:03:59.880 --> 00:04:02.370
if we doubled all of the dimensions.

00:04:02.370 --> 00:04:04.320
How much would that increase the volume?

00:04:04.320 --> 00:04:06.300
Pause the video and see if
you can do that on your own.

00:04:06.300 --> 00:04:09.720
In general, if you double
all the dimensions,

00:04:09.720 --> 00:04:10.860
what does that do to the volume?

00:04:10.860 --> 00:04:12.960
Or if you halve all of the dimensions,

00:04:12.960 --> 00:04:14.793
what does that do to the volume?


Example finding area of triangle

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.880
- [Instructor] What is
the area of the triangle?

00:00:02.880 --> 00:00:04.650
And they're talking
about this triangle here.

00:00:04.650 --> 00:00:06.870
And like always, pause this video

00:00:06.870 --> 00:00:09.120
and see if you can figure
it out on your own.

00:00:10.050 --> 00:00:12.330
All right, now let's
work through it together.

00:00:12.330 --> 00:00:16.500
So we know that the area of a triangle

00:00:16.500 --> 00:00:20.740
is going to be equal to 1/2 times our base

00:00:21.600 --> 00:00:23.403
times our height.

00:00:24.390 --> 00:00:26.430
Let me do the height in a different color.

00:00:26.430 --> 00:00:29.700
Times, times our height.

00:00:29.700 --> 00:00:34.263
So, what is the length of
our base in this scenario?

00:00:35.130 --> 00:00:38.280
Well, the base is this 18 right over here.

00:00:38.280 --> 00:00:39.780
Let me highlight it.

00:00:39.780 --> 00:00:44.340
This length right over here is our base.

00:00:44.340 --> 00:00:47.010
So the base is 18.

00:00:47.010 --> 00:00:48.840
And what is the height?

00:00:48.840 --> 00:00:53.310
Well, the height, we see, is six.

00:00:53.310 --> 00:00:54.420
They give it to us.

00:00:54.420 --> 00:00:55.710
They don't always give it to you,

00:00:55.710 --> 00:00:57.720
but in this example they do.

00:00:57.720 --> 00:01:00.680
So the height is, the height...

00:01:01.650 --> 00:01:06.060
The height is, I'm having
trouble with my pen, is six.

00:01:06.060 --> 00:01:07.410
So now we just have to compute

00:01:07.410 --> 00:01:10.830
what 1/2 times 18 times six is.

00:01:10.830 --> 00:01:12.630
Well, let's think about this a little bit.

00:01:12.630 --> 00:01:15.333
Well, 1/2 times 18 is nine.

00:01:16.230 --> 00:01:20.370
Nine times six is equal to what?

00:01:20.370 --> 00:01:24.393
Nine times six is equal to 54.

00:01:25.440 --> 00:01:26.273
And we are done.

00:01:26.273 --> 00:01:28.773
This is 54 square units.