Calculating the Volume of a Sphere in N-Dimensional Space

Date: 9/5/95 at 0:14:40
From: Anonymous
Subject: geometry 

How do I calculate the volume of a sphere in 4-dimension or, more 
generally, in n-dimension space?

My regards,
Pawel Kowalczyk

Date: 9/5/95 at 13:7:32
From: Doctor Ken
Subject: Re: geometry 

Hello!

This comes from the Frequently Asked Questions of the 
sci.math newsgroup:

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''.

-Doctor Ken,  The Geometry Forum