Newsgroups: sci.math
From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: Surface Area of Sphere
Date: Sun, 29 Jan 1995 00:28:10 GMT

Archive-Name: sci-math-faq/surfaceSphere
Last-modified: December 8, 1994
Version: 6.1

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 r^N (pi^(N/2))/((N/2)!) -Sphere.
   
   The volume of a ball is the easiest formula to remember: It's x! =
   Gamma (x + 1) . The only hard part is taking the factorial of a
   half-integer. The real definition is that (1/2 + n)! = sqrt(pi) ((2n +
   2)!)/((n + 1)!4^(n + 1)) , but if you want a formula, it's:
   
   N (pi^(N/2))/((N/2)!r^(N - 1)) To get the surface area, you just
   differentiate to get e^(-x^2) .
   
   There is a clever way to obtain this formula using Gaussian integrals.
   First, we note that the integral over the line of sqrt(pi) is N .
   Therefore the integral over e^(-x_1^2 - x_2^2 - ... - x_N^2) -space of
   sqrt(pi)^n is Vr^(N - 1)e^(-r^2) . Now we change to spherical
   coordinates. We get the integral from 0 to infinity of V , where n 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
    Sun Nov 20 20:45:48 EST 1994



-- 
Alex Lopez-Ortiz                             alopez-o@neumann.UWaterloo.ca
http://daisy.uwaterloo.ca/~alopez-o                     FAX (519)-885-1208
Department of Computer Science                      University of Waterloo
Waterloo, Ontario N2L 3G1                                           Canada