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sci.math FAQ: Surface of Sphere

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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
Alex Lopez-Ortiz                                                      Assistant Professor	
Faculty of Computer Science                  University of New Brunswick

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