Answer by Kevin P. Costello for What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

A variation on some of the previous arguments that gives some intuition without actually doing any calculation.

Consider $B_n$, the ball in $R^n$, and $C_n$, the cube $[-\frac{1}{2}, \frac{1}{2}]^n$. We make the following observations.

1. $C_n$ has volume $1$.

2. A typical point in $C_n$ will have about half its coordinates larger than $\frac{1}{4}$ in absolute value, so will be outside of $B_n$. In other words, almost none of the volume of $C_n$ is contained in $B_n$.

3. A typical point in $B_n$ will have no coordinates larger than $\frac{1}{2}$, since the sum of the squares of the coordinates is $1$ and this sum has to be divided among $n$ coordinates. (This is a weak version of the concentration of measure mentioned by Gil Kalai, and may be intuitively palatable).

Looking at these, we see that in going from $C_n$ to $B_n$ we start with a volume of $1$, throw almost all of it away, and add almost nothing back in.