The following is a simple and beautiful proof, shown to me to my great delight while I was in high school, that the -ball of radius has -volume

Although I expect most readers will know it, I believe that everybody should see it. I don’t know the history of it, and would be interested in learning.

Suppose that the -ball of radius has -volume . Then, by considering linear transformations, the -ball of radius has -volume . Moreover, differentiating this with respect to should produce the surface -volume of the -sphere : thus we expect .

Now consider the integral

We will compute in two different ways. On the one hand,

On the other hand, the form of suggests introducing a radial coordinate . Computing this way,