Volume of a hypersphere

The volume of a hypersphere with radius $R$ in $n$ dimensions is given by the expression1

\[ V_n(R) = \frac{\pi^{n/2}}{(n/2)!} R^n . \]

We will show this by induction on $n$. The base cases can be checked directly, where we make use of polar coordinates in two dimensions:

V_1(R) = \int_{-R}^R\newcommand{\d}{\,\mathrm{d}}\d x = 2R = \frac{\pi^{1/2}}{(1/2)!} R \\
V_2(R) = \iint\limits_{x_1^2+x_2^2\leq R^2}\d x_2\d x_1 = \int_0^{2\pi}\int_0^R r\d r\d\theta = 2\pi\biggl[\frac{r^2}{2}\biggr]_0^R = \pi R^2

Suppose the formula holds in dimension $n-2$. Using this, we will show that the formula holds in dimension $n$:

V_n(R) &= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}\;\int\limits_{x_1^2+x_2^2+x_3^2\leq R^2}\dotsi\int\limits_{x_1^2+\dotsb+x_n^2\leq R^2}\d x_n\dotsm\d x_1 \\
&= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}\;\int\limits_{x_3^2\leq R^2-x_1^2-x_2^2}\dotsi\int\limits_{x_3^2+\dotsb+x_n^2\leq R^2-x_1^2-x_2^2}\d x_n\dotsm\d x_1 \\
&= \int\limits_{x_1^2\leq R^2}\;\int\limits_{x_1^2+x_2^2\leq R^2}V_{n-2}\Bigl(\sqrt{R^2-x_1^2-x_2^2}\Bigr)\d x_2\d x_1 \\
&= \frac{\pi^{n/2-1}}{(n/2-1)!}\iint\limits_{x_1^2+x_2^2\leq R^2}\sqrt{R^2-x_1^2-x_2^2}^{n-2}\d x_2\d x_1 \\
&= \frac{\pi^{n/2-1}}{(n/2-1)!}\int_0^{2\pi}\int_0^R\sqrt{R^2-r^2}^{n-2}r\d r\d\theta \\
&= \frac{2\pi^{n/2}}{(n/2-1)!}\biggl[-\frac{1}{n}\sqrt{R^2-r^2}^n\biggr]_0^R \\
&= \frac{\pi^{n/2}}{(n/2)!} R^n

By induction, the formula holds for all positive integers $n$.

  1. As one might expect, the factorial with a noninteger argument is simply notation for the gamma function, i.e., $n!:=\Gamma(n+1)$.