You could say you just go round and round hunting for it, but no matter how hard you try you just can’t corner it.

Well, you could.

there’s a way to tie it back to circles

Not necessarily circles, but conic sections. When you take a series of a fixed exponent over a variable x, and graph it, that graph is a parabola.

A parabola is a slice through a cone. Tada, pi appears.