Newsgroups: sci.math,sci.answers,news.answers Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!paperboy.osf.org!bone.think.com!blanket.mitre.org!agate!howland.reston.ans.net!spool.mu.edu!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math FAQ: Surface Area of Sphere Summary: Part 35 of many, New version, Originator: alopez-o@neumann.uwaterloo.ca Message-ID: Sender: news@undergrad.math.uwaterloo.ca (news spool owner) Approved: news-answers-request@MIT.Edu Date: Fri, 17 Nov 1995 17:16:30 GMT Expires: Fri, 8 Dec 1995 09:55:55 GMT Reply-To: alopez-o@neumann.uwaterloo.ca Nntp-Posting-Host: neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math Lines: 45 Xref: senator-bedfellow.mit.edu sci.math:124753 sci.answers:3468 news.answers:58011 Archive-Name: sci-math-faq/surfaceSphere Last-modified: 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 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''. _________________________________________________________________ alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995