I am not sure how appropriate this question is for MO. If it is not, I apologize in advance but I could not resist asking it and if by any chance I get some interesting answers, it will for sure be very useful to keep my students excited about mathematics and physics as September arrives.

We all know very well that $\pi$ (the ratio of the circumference of a circle to its diameter in Euclidean space) is irrational and even transcendental. These are some of the famous results in all mathematics.

So I was wondering what will go wrong if $\pi$ was just an integer number?

Are there important theorems that are based on the fact that it is actually irrational and/or transcendental?