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Formula for the Surface Area of a sphere in Euclidean N-Space

This is equivalent to the volume of the N-1 solid which comprises the boundary of an N-Sphere.

The volume of a ball is the easiest formula to remember: It's r^N (pi^(N/2))/((N/2)!). The only hard part is taking the factorial of a half-integer. The real definition is that x! = Gamma (x + 1), but if you want a formula, it's:

(1/2 + n)! = sqrt(pi) ((2n + 2)!)/((n + 1)!4^(n + 1))

To get the surface area, you just differentiate to get N (pi^(N/2))/((N/2)!)r^(N - 1).

There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of e^(-x^2) is sqrt(pi). Therefore the integral over N-space of e^(-x_1^2 - x_2^2 - ... - x_N^2) is sqrt(pi)^n. Now we change to spherical coordinates. We get the integral from 0 to infinity of Vr^(N - 1)e^(-r^2), where V is the surface volume of a sphere. Integrate by parts repeatedly to get the desired formula.

It is possible to derive the volume of the sphere from ``first principles''.

Alex Lopez-Ortiz
Fri Feb 20 21:45:30 EST 1998