Is there any way to prove for any n-sided polygon that if it's regular, it has the highest possible area/circumference ratio?

I have been thinking about this for days, and I can't find an answer. I have thought about calculus, but the problem is that there are a variable number of variables. You end up with a formula containing sums, which you can't take the derivative of (that's true, is it?)

Maybe with a geometric approach? But I have no idea how to do that...