Surface Area of an n-dimensional Sphere

Date: 07/28/97 at 18:32:35
From: Christine Bollinger
Subject: Surface area of an n dimensional sphere

I was wondering how to calculate the surface area of a sphere in n 
dimensions.  I read and understand your explanation on how to 
calculate the VOLUME of an n-dimensional sphere and know that the two 
are related, but I can't figure out how.  

Thanks in advance,

Christine Bollinger

Date: 07/31/97 at 11:44:17
From: Doctor Rob
Subject: Re: Surface area of an n dimensional sphere

Take a look at this Web page with the Formula for the Surface Area of 
a sphere in Euclidean N-Space:

  http://daisy.uwaterloo.ca/~alopez-o/math-faq/node42.html   

The relation is that the measure of the surface of an n-sphere is the 
derivative of the volume of the n-sphere with respect to the radius.  
If you know enough calculus to figure that out, that is your answer.  
If not, this is how that is computed.

Let c = pi^(n/2)/((n/2)!), then

V = c*r^n.

If we increase the radius to r + a, then we get 

V' = c*(r+a)^n.

The difference in volume is V' - V, and the difference in volume per
unit increase in radius is

   (V' - V)/((r + a) - r)
     = c*((r+a)^n - r^n)/a

Use the Binomial Theorem to expand (r+a)^n, and notice that the first
term is canceled by the -r^n term above.  All the other terms are
divisible by a, so we divide the denominator a into them. The result
is
     = c*C(n,1)*r^(n-1) + c*C(n,2)*r^(n-2)*a + ...

Now we consider what happens as a gets very small.  All the terms
from the second one on get very small, too.  If we actually put a = 0,
we get the formula

   S = c*n*r^(n-1).

The formula for surface area is then just S = V*n/r.

I hope this is what you wanted.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/