The volume of an n-ball is
\begin{align}\dfrac{\pi^{n/2}r^n}{\Gamma\left(n/2+1\right)}\end{align}
However if we define $\alpha = \sqrt{\pi}$, and use the $\Pi$-function, we get
\begin{align}\dfrac{\left(\alpha r\right)^n}{\Pi(n/2)}\end{align}
which is simpler. However, this form isn't very useful, as it's not very easy to calculate $\Pi()$ of half integers.
Another formula, given $\eta = \pi/2$ and the diameter $d = 2r$, is
\begin{align}\dfrac{d^n\eta^{\lfloor n/2\rfloor}}{n!!}\end{align}
This is simpler to calculate, but it's not as pretty as it uses the diameter. Both of these functions also uses half integers; are there any functions without half integers and using $r$ which are simple to calculate?