List of formulas in elementary geometry

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

Sources:

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:

• 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.

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:

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

• 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