volume of the $n$-sphere

The volume contained inside $S^{n}$, the $n$-sphere (or hypersphere), is given by the integral

 $V(n)=\int_{\sum_{i=1}^{n+1}x_{i}^{2}\leq 1}d^{n+1}x.$

Going to polar coordinates ($r^{2}=\sum_{i=1}^{n+1}x_{i}^{2}$) this becomes

 $V(n)=\int_{S^{n}}d\Omega\int_{0}^{1}r^{n}\,dr.$

The first integral is the integral over all solid angles subtended by the sphere and is equal to its area $A(n)=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}$, where $\Gamma(x)$ is the gamma function. The second integral is elementary and evaluates to $\int_{0}^{1}r^{n}\,dr=1/(n+1)$.

Finally, the volume is

 $V(n)=\frac{\pi^{\frac{n+1}{2}}}{\frac{n+1}{2}\Gamma\left(\frac{n+1}{2}\right)}% =\frac{\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+3}{2}\right)}.$

If the sphere has radius $R$ instead of $1$, then the correct volume is $V(n)R^{n+1}$.

Note that this formula works for $n\geq 0$. The first few cases are

• $n=0$

$\Gamma(3/2)=\sqrt{\pi}/2$, hence $V(0)=2$ (this is the length of the interval $[-1,1]$ in $\mathbb{R}$);

• $n=1$

$\Gamma(2)=1$, hence $V(1)=\pi$ (this is the familiar result for the area of the unit circle);

• $n=2$

$\Gamma(5/2)=3\sqrt{\pi}/4$, hence $V(2)=4\pi/3$ (this is the familiar result for the volume of the sphere);

• $n=3$

$\Gamma(3)=2$, hence $V(3)=\pi^{2}/2$.

Title volume of the $n$-sphere VolumeOfTheNsphere 2013-03-22 13:47:09 2013-03-22 13:47:09 CWoo (3771) CWoo (3771) 10 CWoo (3771) Derivation msc 51M05 AreaOfTheNSphere