Has anyone come across a geometric interpretation for fractional exponents of pi? Once in a while I'll see pi, not squared, but to a fractional power. For instance when dealing with a bell curve with its integral to infinity, you obtain $$ \frac{\sqrt{\pi}}{2}$$
When you evaluate certain elliptic integrals or fractional inputs of the gamma function like $\Gamma(\frac{2}{3})$, you might obtain a $$\pi^{\frac{2}{3}}$$
Getry: in Sterling's formula which is $$n! \sim \sqrt{2 \pi n} (\frac{n}{e})^n$$ you can see the square root of pi. 
But, where do fractional powers of pi occur geometrically? In what physical circumstances do these uncommon numbers like this typically occur? If I drew a circle...where is that theorem containing a fractional power of pi that relates its circumference to its diameter? Or maybe it's not a circle, maybe it pertains to a lemiscate, or maybe it pertains to  ellipse, but there has to be some something that can make sense of these numbers, they aren't random. 

To generalize Professor Vector's comment, the $d$-dimensional hypersphere of radius 1 has hypervolume $A_d\pi^s$ for some easily computed rational constant $A_d$, where $s$ is the integer part of $d/2$ – see Wikipedia. Then the hypercube with the same hypervolume has side $A_d^{1/d}\pi^{s/d}$
