# Volume of n-dimensional sphere

1. Nov 20, 2013

### Superposed_Cat

Hello, this may seem like a stupid question but how would one calculate the volume of an n-dimensional sphere?
Thanks.

2. Nov 20, 2013

### Staff: Mentor

I think you would extroplate the formulas for area of a circle to volume of a sphere to hypervolume of a hypersphere...

pi*r^2
4/3pi*r^3

http://en.wikipedia.org/wiki/N-sphere

Midway through the article is a cool table of the progress of n from 0 to ... and the volumes and surfaces of the hyperspheres.

3. Nov 21, 2013

### JJacquelin

4. Nov 21, 2013

### HallsofIvy

Given an $x_1,x_2, ..., x_n$ Cartesian coordinate system, the the equation of the n-sphere of radius R, with center at the origin is $x_1^2+ x_2^2+ x_3^2+ \cdot\cdot\cdot+ x_n^2= R^2$.
It is clear that, if all the other variables are 0, then $x_1^2= R^2$ so that $x_1$ ranges between -R and R to cover the entire n-sphere. In the $x_1x_2$ plane, all other variables 0, $x_1^2+ x_2^2= R^2$ so that, for fixed $x_1$, $x_2= \pm\sqrt{R^2- x_1^2}$ and so $x_2$ ranges between $-\sqrt{R^2- x_1^2}$ and $\sqrt{R^2- x_1^2}$ etc.

Continuing like that, we see that the volume is given by
$$\int_{-R}^R\int_{-\sqrt{R^2- x_1^2}}^{\sqrt{R^2- x_1^2}}\int_{-\sqrt{R^2- x_1^2- x_2^2}}^{\sqrt{R^2- x_1^2- x_2^2}}\cdot\cdot\cdot\int_{-\sqrt{R^2- x_1^2- x_2^2- \cdot\cdot\cdot- x_{n-1}^2}}^{\sqrt{R^2- x_1^2- x_2^2- \cdot\cdot\cdot- x_{n-1}^2}} dx_ndx_{n-1}\cdot\cdot\cdot dx_2 dx_1$$.

You ought to be able to take the formulas for area of a circle (2-sphere), volume of a sphere (3-sphere) and use that integral to find the hyper-volumes of the 4-sphere, 5-sphere, etc to find a general formula.

5. Nov 21, 2013

### mathman

http://en.wikipedia.org/wiki/N-sphere

Above is a detailed discussion. Note that the development works with a generalization of spherical coordinates.

6. Nov 22, 2013

### Superposed_Cat

Thanks all especially halls of ivy as I was just about to ask for it in integral form