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```Archive-Name: sci-math-faq/surfaceSphere
Last-modified: December 8, 1994
Version: 6.2

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''.

_________________________________________________________________

alopez-o@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995

```