Lebesgue Measure of a ball in $\mathbb{R}^n$

Question

What is the relationship between the Lebesgue measure of a ball in $\mathbb{R}^n$ to the measure of a sphere? I've derived the measure of a sphere $S^{n-1}$ in $\mathbb{R}^n$ to be $\frac{2\pi^{n/2}}{\Gamma(n/2)}$, but I don't know how to relate the two. Please help and thanks in advance!

Answer 1

The area of a sphere of radius $r$ is the derivative with respect to $r$ of the volume of the ball of radius $r$. (Think about the volume of a thin film of thickness $dr$ on the surface of the ball!) Since that volume is proportional to $r^n$, it follows that the area of the unit sphere is $n$ times the volume of the unit ball.