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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
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
```