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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 . The only hard part is taking the factorial of a half-integer. The real definition is that , but if you want a formula, it's:

To get the surface area, you just differentiate to get .

There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of is . Therefore the integral over N-space of is . Now we change to spherical coordinates. We get the integral from 0 to infinity of , 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
Mon Feb 23 16:26:48 EST 1998