List of formulas in elementary geometry
# List of formulas in elementary geometry

This is a short list of some common mathematical shapes and figures and the formulas that describe them.

## Two-dimensional shapes

| Shape | Area | Perimeter/Circumference | Meanings of symbols |
| - | - | - | - |
| Square | $l^{2}$ | $4l$ | $l$ is the length of a side |
| Rectangle | $lb$ | $2(l+b)$ | $l$ is length, $b$ is breadth |
| Circle | $\pi r^{2}$ | $2\pi r$ or $\pi d$ | where $r$ is the radius and $d$ is the diameter |
| Ellipse | $\pi ab$ | | where $a$ is the semimajor axis and $b$ is the semiminor axis |
| Triangle | ${\frac {bh}{2}}$ | $a+b+c$ | $b$ is base; $h$ is height; $a,b,c$ are sides |
| Parallelogram | $bh$ | $2(a+b)$ | $b$ is base, $h$ is height, $a$ is side |
| Trapezoid | ${\frac {a+b}{2}}h$ | | $a$ and $b$ are the bases |

    Sources:

## Three-dimensional shapes

This is a list of volume formulas of basic $shapes:^{:405–406}$

* Cone – 1/3 $πr^2h{\textstyle$ {\frac {1}{3}}\pi r^{2}h}, where r{\textstyle r} is the base's radius
* Cube – $a^3{\textstyle$ a^{3}}, where a{\textstyle a} is the side's length;
* Cuboid – abc{\textstyle abc}, where a{\textstyle a}, b{\textstyle b}, and c{\textstyle c} are the sides' length;
* Cylinder – $πr^2h{\textstyle$ \pi r^{2}h}, where r{\textstyle r} is the base's radius and h{\textstyle h} is the cone's height;
* Ellipsoid – 4/3 πabc{\textstyle {\frac {4}{3}}\pi abc}, where a{\textstyle a}, b{\textstyle b}, and c{\textstyle c} are the semi-major and semi-minor axes' length;
* Sphere – 4/3 $πr^3{\textstyle$ {\frac {4}{3}}\pi r^{3}}, where r{\textstyle r} is the radius;
* Parallelepiped – abc√K{\textstyle abc{\sqrt {K}}}, where a{\textstyle a}, b{\textstyle b}, and c{\textstyle c} are the sides' length, $K=1+2cos⁡(α)cos⁡(β)cos⁡(γ)−cos^2⁡(α)−cos^2⁡(β)−cos^2⁡(γ){\textstyle$ K=1+2\cos(\alpha )\cos(\beta )\cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}, and α{\textstyle \alpha }, β{\textstyle \beta }, and γ{\textstyle \gamma } are angles between the two sides;
* Prism – Bh{\textstyle Bh}, where B{\textstyle B} is the base's area and h{\textstyle h} is the prism's height;
* Pyramid – 1/3 Bh{\textstyle {\frac {1}{3}}Bh}, where B{\textstyle B} is the base's area and h{\textstyle h} is the pyramid's height;
* Tetrahedron – √2/12 $a^3{\textstyle$ {{\sqrt {2}} \over 12}a^{3}}, where a{\textstyle a} is the side's length.

### Sphere

The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables

* $r$ is the radius,
* $C=2\pi r$ is the circumference (the length of any one of its great circles),
* $S$ is the surface area,
* $V$ is the volume.

Surface area:

${\begin{alignedat}{4}S&=4\pi r^{2}\\[0.3ex]&={\frac {1}{\pi }}C^{2}\\[0.3ex]&={\sqrt[{3}]{\pi (6V)^{2}}}\\[0.3ex]\end{alignedat}}$

Volume:

${\begin{alignedat}{4}V&={\frac {4}{3}}\pi r^{3}\\[0.3ex]&={\frac {1}{6\pi ^{2}}}C^{3}\\[0.3ex]&={\frac {1}{6{\sqrt {\pi }}}}S^{3/2}\\[0.3ex]\end{alignedat}}$

Radius:

${\begin{alignedat}{4}r&={\frac {1}{2\pi }}C\\[0.3ex]&={\sqrt {{\frac {1}{4\pi }}S}}\\[0.3ex]&={\sqrt[{3}]{{\frac {3}{4\pi }}V}}\\[0.3ex]\end{alignedat}}$

Circumference:

${\begin{alignedat}{4}C&=2\pi r\\[0.3ex]&={\sqrt {\pi S}}\\[0.3ex]&={\sqrt[{3}]{\pi ^{2}6V}}\\[0.3ex]\end{alignedat}}$

