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

Sources:^[1][2][3]

Three-dimensional shapes

Illustration of the shapes' equation terms

Cube

Cuboid

Prism

Parallelepiped

Pyramids

Tetrahedron

Cone

Cylinder

Sphere

Ellipsoid

This is a list of volume formulas of basic shapes:^{[4]: 405–406}

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

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

Surface area:

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

Volume:

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

Radius:

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

Circumference:

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

See also

• Arc length – Distance along a curve
• Area#Area formulas – Size of a two-dimensional surface
• Perimeter#Formulas – Path that surrounds an area
• List of second moments of area
• List of surface-area-to-volume ratios – Surface area per unit volume
• List of surface area formulas – Measure of a two-dimensional surface
• List of trigonometric identities – Equalities that involve trigonometric functions
• List of volume formulas – Quantity of three-dimensional space

References

This page was last edited on 16 June 2023, at 16:58