See reader questions & answers on this topic!  Help others by sharing your knowledge Archivename: scimathfaq/surfaceareaofsphere Lastmodified: February 20, 1998 Version: 7.5 Formula for the Surface Area of a sphere in Euclidean NSpace This is equivalent to the volume of the N1 solid which comprises the boundary of an NSphere. 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 Nspace 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 LopezOrtiz alopezo@unb.ca http://www.cs.unb.ca/~alopezo Assistant Professor Faculty of Computer Science University of New Brunswick User Contributions:
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Last Update March 27 2014 @ 02:12 PM

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