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''.