How do I prove these general formulas for the volume of the n-sphere?

**Kristenx3** · December 6th 2012, 05:00 AM

Hi, I am working on a project dealing with n-spheres.

I have an even equation
Vn=(1/n!)*(pi^n)*(R^2n)

and an odd equation
Vn=((2^(2n+1)n!)/(2n+1)!)*(pi^n)*R^2n+1

How can I prove these? I have no idea how to arrive at them. We have one general formula,
Vn(R)=2(integral from 0 to pi/2)Vn-1(Rcos(theta))Rcos(theta)d(theta)

**HallsofIvy** · December 6th 2012, 08:31 AM

Well, my first thought is to actually do the integration in that "general formula"!

However, that will not give you the correct answer because your "general formula" is wrong. Where did you get it?

**SworD** · December 6th 2012, 08:56 AM

This derivation might not be as rigorous as you want, depending on the context. But here it goes:

The volume of an n-sphere will be proportional to r^n. (For example, volume of 3-sphere is 4/3*Pi*r^3).

Let Cn be the proportionality constant, so that:

$V = C_n r^n$

The first step is to recognize through a geometric argument the following fact:

$C_{n+1} = \int_{-1}^{1} C_n (\sqrt{1-x^2})^n dx$

Therefore:

$\frac{C_{n+1}}{C_n} = \int_{-1}^{1} (\sqrt{1-x^2})^n dx$

When you make the substitution $x = \sin{\theta}$

This integral becomes:

$\frac{C_{n+1}}{C_n} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{\theta})^{n+1} d\theta$

or

$\frac{C_{n}}{C_{n-1}} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{\theta})^{n} d\theta$

You can see an evaluation of the integral in this thread:
Evaluate the integral

Which leads us to

$\frac{C_{n}}{C_{n-1}} = \frac{\sqrt{\pi}\Gamma((1+n)/2)}{\Gamma(1+n/2)}$

So the goal is to find a formula for C_n and the initial condition that C_1 = 2. Any such function will work. It can be quickly shown that this formula holds:

$C_n = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}$

If you are unfamiliar with the gamma function, read about it here.

Using that formula for C_n, you can break it down into the special odd and even cases using the facts that

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$

and

$\Gamma(x) = (x-1)!$

You should check out the formulas on this page as well, it has other ways to derive it:

n-sphere - Wikipedia, the free encyclopedia

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