# Sphere

round, rotationally symmetric shape of the 2D surface of a ball in 3D space

A sphere is a round, three-dimensional shape. All points on the edge of the sphere are at the same distance from the center. The distance from the center is called the radius of the sphere.

Common things that have the shape of a sphere are basketballs, superballs, DragonBalls and playground balls. The Earth and the Sun are nearly spherical, meaning sphere-shaped.

A sphere is the three-dimensional analog of a circle.

## Calculating measures of a sphere

### Surface area

Using the circumference: ${\displaystyle A={\frac {c^{2}}{\pi }}={\frac {2c^{2}}{\tau }}}$

Using the diameter: ${\displaystyle A=\pi d^{2}={\frac {\tau d^{2}}{2}}}$

Using the radius: ${\displaystyle A=2\tau r^{2}=4\pi r^{2}}$

Using the volume: ${\displaystyle A={\sqrt[{3}]{3\tau V^{2}}}={\sqrt[{3}]{6\pi V^{2}}}}$

### Circumference

Using the surface area: ${\displaystyle c={\sqrt {\pi A}}={\sqrt {\frac {\tau A}{2}}}}$

Using the diameter: ${\displaystyle c=\pi d={\frac {\tau d}{2}}}$

Using the radius: ${\displaystyle c=\tau r=2\pi r}$

Using the volume: ${\displaystyle c={\sqrt[{3}]{6\pi ^{2}V}}={\sqrt[{3}]{\frac {3\tau ^{2}V}{2}}}}$

### Diameter

Using the surface area: ${\displaystyle d={\sqrt {\frac {A}{\pi }}}={\sqrt {\frac {2A}{\tau }}}}$

Using the circumference: ${\displaystyle d={\frac {c}{\pi }}={\frac {2c}{\tau }}}$

Using the radius: ${\displaystyle d=2r}$

Using the volume: ${\displaystyle d={\sqrt[{3}]{\frac {6V}{\pi }}}={\sqrt[{3}]{\frac {12V}{\tau }}}}$

Using the surface area: ${\displaystyle r={\sqrt {\frac {A}{2\tau }}}={\sqrt {\frac {A}{4\pi }}}}$

Using the circumference: ${\displaystyle r={\frac {c}{\tau }}={\frac {c}{2\pi }}}$

Using the diameter: ${\displaystyle r={\frac {d}{2}}}$

Using the volume: ${\displaystyle r={\sqrt[{3}]{\frac {3V}{2\tau }}}={\sqrt[{3}]{\frac {3V}{4\pi }}}}$

### Volume

Using the surface area: ${\displaystyle V={\sqrt {\frac {A^{3}}{18\tau }}}={\sqrt {\frac {A^{3}}{36\pi }}}}$

Using the circumference: ${\displaystyle V={\frac {c^{3}}{6\pi ^{2}}}={\frac {2c^{3}}{3\tau ^{2}}}}$

Using the diameter: ${\displaystyle V={\frac {\pi d^{3}}{6}}={\frac {\tau d^{3}}{12}}}$

Using the radius: ${\displaystyle V={\frac {2\tau r^{3}}{3}}={\frac {4\pi r^{3}}{3}}}$

## Equation of a sphere

In Cartesian coordinates, the equation for a sphere with a center at ${\displaystyle (x_{0},y_{0},z_{0})}$  is as follows:

${\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}}$

where ${\displaystyle r}$  is the radius of the sphere.