# Sphere

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

A sphere is a shape in space that is like the surface of a ball. Usually, the words ball and sphere mean the same thing. But in mathematics, a sphere is the surface of a ball, which is given by all the points in three dimensional space that are located at a fixed distance from the center. The distance from the center is called the radius of the sphere. When the sphere is filled in with all the points inside, it is called a ball.

Common things that have the shape of a sphere are basketballs, superballs, 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: $A={\frac {c^{2}}{\pi }}={\frac {2c^{2}}{\tau }}$

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

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

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

### Circumference

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

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

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

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

### Diameter

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

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

Using the radius: $d=2r$

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

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

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

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

Using the volume: $r={\sqrt[{3}]{\frac {3V}{2\tau }}}={\sqrt[{3}]{\frac {3V}{4\pi }}}\approx {\frac {\sqrt[{3}]{15V}}{4}}$  (more simple but less precise)

### Volume

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

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

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

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

## Equation of a sphere

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

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

where $r$  is the radius of the sphere.