See reader questions & answers on this topic!  Help others by sharing your knowledge ArchiveName: scimathfaq/surfaceSphere Lastmodified: 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 halfinteger. 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''. _________________________________________________________________ alopezo@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 User Contributions:
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