Volume of a Hypersphere

Lemma:

$V_N(R) \propto R^N$ where $V_N$ is volume of N-dim hypersphere.
The proportionality constant is $V_N(1)$, so $V_N(R) = V_N(1) \times R^N$.

Recursive formula:

$V_N(R) = \frac{2 \pi R^2}{N} \times V_{N-2}(R)$

Random: A 0-dim hypersphere has volume 1.