Volume of an n-ball

In geometry, a ball is a region in space comprising all points within a fixed distance from a fixed point. An $n$ -ball is a ball in $n$ -dimensional Euclidean space. The volume of an $n$ -ball is an important constant that occurs in formulas throughout mathematics.

The $n$ -dimensional volume of a Euclidean ball of radius $R$ in $n$ -dimensional Euclidean space is:

where $Γ$ is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function to noninteger arguments). Using explicit formulas for particular values of the gamma function at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are:

In the formula for odd-dimensional volumes, the double factorial $(2 k + 1)!!$ is defined for odd integers $2 k + 1$ as $(2 k + 1)!! = 1 \cdot 3 \cdot 5 \cdot \dots \cdot (2 k - 1) \cdot (2 k + 1)$ .

Instead of expressing the volume $V$ of the ball in terms of its radius $R$ , the formula can be inverted to express the radius as a function of the volume:

This formula, too, can be separated into even- and odd-dimensional cases using factorials and double factorials in place of the gamma function:

The volume satisfies several recursive formulas. These formulas can either be proved directly or proved as consequences of the general volume formula above. The simplest to state is a formula for the volume of an $n$ -ball in terms of the volume of an $(n - 2)$ -ball of the same radius:

There is also a formula for the volume of an $n$ -ball in terms of the volume of an $(n - 1)$ -ball of the same radius:

Using explicit formulas for the gamma function again shows that the one-dimension recursion formula can also be written as:

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