In geometry, a ball is a region comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n-ball, the n-ball of radius 1. The real number V n {\displaystyle V_{n}} can be expressed via a two-dimension recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms of A n {\displaystyle A_{n}} , the area of the unit n-sphere.
Formulas
The first volumes are as follows:
Closed form The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:
V n ( R ) = π n / 2 Γ ( n 2 + 1 ) R n , {\displaystyle V_{n}(R)={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}R^{n},}
where Γ is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies Γ(n) = (n − 1)! if n is a positive integer and Γ(n + 1/2) = (n − 1/2) · (n − 3/2) · … · 1/2 · π1/2 if n is a non-negative integer.
Two-dimension recurrence relation The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
V n ( R ) = { 1 if n = 0 , 2 R if n = 1 , 2 π n R 2 × V n − 2 ( R ) otherwise . {\displaystyle V_{n}(R)={\begin{cases}1&{\text{if }}n=0,\\[0.5ex]2R&{\text{if }}n=1,\\[0.5ex]{\dfrac {2\pi }{n}}R^{2}\times V_{n-2}(R)&{\text{otherwise}}.\end{cases}}}
This allows computation of Vn(R) in approximately n / 2 steps.
Alternative forms The volume can also be expressed in terms of an (n − 1)-ball using the one-dimension recurrence relation:
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