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Subject: sci.math FAQ: Surface of Sphere

This article was archived around: 17 Feb 2000 22:55:54 GMT

All FAQs in Directory: sci-math-faq
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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 alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick