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

Illustration of the shapes' equation terms

File:Wuerfel-1-tab.svg

Cube

File:Quader-1-tab.svg

Cuboid

File:Prisma-1-e.svg

Prism

File:Parallelepiped-1-tab.svg

Parallelepiped

File:Pyramide-46-e.svg

Pyramids

File:Tetraeder-1-tab.svg

Tetrahedron

File:Kegel-1-tab.svg

Cone

File:Zylinder-1-tab.svg

Cylinder

File:Kugel-1-tab.svg

Sphere

File:Ellipsoid-1-tab.svg

Ellipsoid

This is a list of volume formulas of basic shapes:

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

Sphere

See also: Volume of an n-ball and n-sphere § Volume and surface area

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: $$\begin{array}{rl}S& =4\pi r^2\\ & =\frac{1}{\pi }{C}^2\\ & =\sqrt[3]{\pi (6V{)}^2}\\ \end{array}$$ Volume: $$\begin{array}{rl}V& =\frac{4}{3}\pi r^3\\ & =\frac{1}{6{\pi }^2}{C}^3\\ & =\frac{1}{6\sqrt{\pi }}{S}^{3/2}\\ \end{array}$$ Radius: $$\begin{array}{rl}r& =\frac{1}{2\pi }C\\ & =\sqrt{\frac{1}{4\pi }S}\\ & =\sqrt[3]{\frac{3}{4\pi }V}\\ \end{array}$$ Circumference: $$\begin{array}{rl}C& =2\pi r\\ & =\sqrt{\pi S}\\ & =\sqrt[3]{{\pi }^{2}6V}\\ \end{array}$$

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
• List of volume formulas – Quantity of three-dimensional space