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Sphere/N-Sphere facts

The volume of an n-dimensional sphere with radius r is:

$\begin{displaymath} V = \frac{ \pi^{\frac{n}{2}} r^n}{\Gamma(\frac{n}{2} + 1)} \nonumber \end{displaymath}$

One obtains the surface area by differentiating the volume with respect to the radius, i.e.

$\begin{displaymath} A = \frac{d V}{d r} = \frac{n \pi^{\frac{n}{2}} r^{n-1}}{\Gamma(\frac{n}{2} + 1)} \nonumber \end{displaymath}$

Note that, $\Gamma(x+1) = x \Gamma(x)$ and that $\Gamma(0.5) = \sqrt{\pi}$