A polyhedron with two congruent and parallel faces and whose lateral faces are parallelograms. Prism are often distinguished by the shape of their base polygon. A prism with rectangular bases is a rectangular prism. A rectangular prism has two ends and four sides.

## What is a Rectangular Prism

A prism with rectangular bases is a rectangular prism. Rectangular prism is a 3-dimensional solid object which has six faces that are rectangles. A rectangular prism has two ends and four sides. Opposite sides of the rectangular prism have the same area.

Rectangular Prism Vertices :
A vertex is the point on 3-D shape where two or more edges meet. More than one vertex contained by a object is called vertices. Rectangular prism have eight vertices.

## Volume of a Rectangular Prism

Volume of the right rectangular prism, $V$, is the product of the, $A$, area of base and, $h$, height of the prism. The base of the rectangular prism is, rectangle. So the area of the base is product of length and breadth or width. The formula for finding the volume of right rectangular prism is $V$ = $lwh$.

Formula for Volume of a Rectangular Prism

The formula for the volume of any right prism,

$\Rightarrow\ V$ = $Ah$

Where, $A$ = Area of the base and $h$ = Perpendicular height

Now

If prism is a rectangular prism.

$\Rightarrow$ Area of the base = $lw$

$\Rightarrow\ V$ = $A \times h$

$\Rightarrow\ lw \times h$

$\Rightarrow\ V$ = $lwh$

The volume of a rectangular prism is found by the formula, $V$ = $lwh$.
Formula:

Volume of a rectangular prism = $lwh$.

Where,

'$l$', '$w$', '$h$' are the length, width and height of the prism.

## Surface Area of a Rectangular Prism

To find the surface area of a rectangular prism, firstly find the area of each side of the rectangular prism. Rectangular prism have six surface area, but there are three sets of two equal areas. The top and bottom, the left side and right side and the front and back will be the same. Length, width and height are three dimensions of rectangular prism, helps to find the surface area. The surface area of the right rectangular prism is $2(lh + hb + bl)$.

Surface Area of Rectangular Prism Formula:

The surface area of a rectangular prism is found by adding the area of all the faces of the prism. In rectangular prism at least two faces will have the same area.

$\Rightarrow$ Surface Area = $2 \times\ Area\ of Front\ +\ 2\ \times\ Area\ of\ Side\ +\ 2\ \times\ Area\ of\ Base$

$\Rightarrow\ 2(lh)\ +\ 2(bh)\ +\ 2(bl)$

$\Rightarrow\ 2(lh\ +\ hb\ +\ bl)$

Formula:

Surface Area of Rectangular Prism = 2(lh + hb + bl).

Where, 'l' is length, 'b' is breadth and 'h' is the height of the rectangular prism.

## Lateral Area of a Rectangular Prism

The lateral surface area is the area of any object with non-base faces. The lateral surface area L of any right rectangular prism is equal to the perimeter of the base times the height of the prism.

$\Rightarrow\ L$ = $Ph$

Where, $P$ is the perimeter of a base and h be the height of the prism.

Now
Perimeter of the rectangular prism is,

$P$ = $2(l + b)$

Where, '$l$' and '$b$' be the length and breadth of the prism.

Formula:

Lateral Surface Area of Rectangular Prism = $P$h = $2h(l + b)$.

Where, '$l$', '$b$' and '$h$' be the length, breadth and height of the prism.

## Rectangular Prism Examples

Below you could see some examples of rectangular prism:
Example 1:

Find the lateral surface area of a box in the shape of prism whose bases are rectangular with side length $10$ cm and $12$ cm, and whose height is $5$ cm.

Solution:

Given:
Dimensions of rectangular prism are,
Length $(l)$ = $10$ cm
Breadth $(b)$ = $12$ cm
Height $(h)$ = $5$ cm

Step 1:

Perimeter of the rectangular prism is,

$P$ = $2(l + b)$

$\Rightarrow\ P$ = $2(10 + 12)$

= $2(22)$

= $44$

$\Rightarrow\ P$ = $44$ cm

Step 2:

Lateral Surface Area of Rectangular Prism = $Ph$ = $2h(l + b)$

$\Rightarrow\ LSA$ = $44 \times\ 5$

= $220$

Hence the lateral surface area of a box is $220\ cm^{2}$.
Example 2:

Find the volume of a rectangular solid with length $9$, width $6$, and height $5$.

Solution:

Given:
Dimensions of rectangular prism are,
Length $(l)$ = $9$
Width $(w)$ = $6$
Height $(h)$ = $5$

Step 1:

Area of the base of the prism,

Therefore, Area of the base = $l \times\ w$

$\Rightarrow\ A$ = $9 \times 6$

= $54$

$\Rightarrow\ A$ = $54$ square units

Step 2:

Volume of a rectangular prism = $Ah$

$\Rightarrow\ V$ = $54 \times 5$

= $270$

Hence the volume of the rectangular prism is $270$ square units.