Is there a geometric proof for irrationality of $\pi$? That would be neat.
There is the beautiful short proof by Ivan Niven, A simple proof that $\pi$ is irrational, by elementary calculus. I am not aware of a pure geometric proof, but this does not mean, of course, that there might not be one. But it seems that the methods needed to prove the irrationality of $\pi$ always require also calculus, see here.