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From: [email protected] (Alex Lopez-Ortiz)
Newsgroups: sci.math,news.answers,sci.answers
Subject: sci.math FAQ: Surface of Sphere
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Date: 17 Feb 2000 22:55:54 GMT
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Archive-name: sci-math-faq/surfaceareaofsphere
Last-modified: February 20, 1998
Version: 7.5

       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''.
    _________________________________________________________________

--
Alex Lopez-Ortiz                                         [email protected]
http://www.cs.unb.ca/~alopez-o                       Assistant Professor
Faculty of Computer Science                  University of New Brunswick