Volume and area

Graphs of volumes (V) and surface areas (S) of unit n-balls.

The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the ${\displaystyle x}$-, ${\displaystyle y}$-, or ${\displaystyle z}$- axes:

${\displaystyle x^{2}+y^{2}+z^{2}=1}$

The volume of the unit ball in Euclidean ${\displaystyle n}$-space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit ${\displaystyle n}$-ball, which we denote ${\displaystyle V_{n},}$ can be expressed by making use of the gamma function. It is

${\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}}$

where ${\displaystyle n!!}$ is the double factorial.

The hypervolume of the ${\displaystyle (n-1)}$-dimensional unit sphere (i.e., the "area" of the boundary of the ${\displaystyle n}$-dimensional unit ball), which we denote ${\displaystyle A_{n-1},}$ can be expressed as

${\displaystyle A_{n-1}=nV_{n}={\frac {n\pi ^{n/2}}{\Gamma (1+n/2)}}={\frac {2\pi ^{n/2}}{\Gamma (n/2)}}={\begin{cases}{2\pi ^{n/2}}/{(n/2-1)!}&\mathrm {if~} n\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}}$

For example, ${\displaystyle A_{0}=2}$ is the "area" of the boundary of the unit ball ${\displaystyle [-1,1]\subset \mathbb {R} }$, which simply counts the two points. Then ${\displaystyle A_{1}=2\pi }$ is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. ${\displaystyle A_{2}=4\pi }$ is the area of the boundary of the unit ball ${\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}}$, which is the surface area of the unit sphere ${\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}$.

The surface areas and the volumes for some values of ${\displaystyle n}$ are as follows:

${\displaystyle n}$	${\displaystyle A_{n-1}}$ (surface area)		${\displaystyle V_{n}}$ (volume)
0			${\displaystyle (1/0!)\pi ^{0}}$	1
1	${\displaystyle 1(2^{1}/1!!)\pi ^{0}}$	2	${\displaystyle (2^{1}/1!!)\pi ^{0}}$	2
2	${\displaystyle 2(1/1!)\pi ^{1}=2\pi }$	6.283	${\displaystyle (1/1!)\pi ^{1}=\pi }$	3.141
3	${\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }$	12.57	${\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }$	4.189
4	${\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}$	19.74	${\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}$	4.935
5	${\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}$	26.32	${\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}$	5.264
6	${\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}$	31.01	${\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}$	5.168
7	${\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}$	33.07	${\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}$	4.725
8	${\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}$	32.47	${\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}$	4.059
9	${\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}$	29.69	${\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}$	3.299
10	${\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}$	25.50	${\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}$	2.550

where the decimal expanded values for ${\displaystyle n\geq 2}$ are rounded to the displayed precision.

Metric spaces

All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.

Quadratic forms

If ${\displaystyle V}$ is a linear space with a real quadratic form ${\displaystyle F:V\to \mathbb {R} ,}$ then ${\displaystyle \{p\in V:F(p)=1\}}$ may be called the unit sphere^[3][4] or unit quasi-sphere of ${\displaystyle V.}$ For example, the quadratic form ${\displaystyle x^{2}-y^{2}}$, when set equal to one, produces the unit hyperbola, which plays the role of the "unit circle" in the plane of split-complex numbers. Similarly, the quadratic form ${\displaystyle x^{2}}$ yields a pair of lines for the unit sphere in the dual number plane.