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Sphere/N-Sphere facts

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

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

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

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

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

Dr. Everett (Skip) F. Carter Jr.
Taygeta Scientific Inc.
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Monterey, CA. 93940-6140
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