What’s the formula for the volume of a 4-dimensional sphere? If you have that one, can you come up with a formula for the volume of an n-dimensional sphere?

Don’t look it up! It’s a good problem. I highly encourage you to work on it before looking at my solution.

More specifically, try to come up with an equation which relates the volume of an n-dimensional sphere to an (n-1) dimensional sphere. You may not be able to analytically evaluate your equation yourself (I wasn’t), but it should be something that a computer could solve.

Click to show solution