Are there any (interesting) consequences of the irrationality of π? 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?  

 A: Since Euler showed that $$\frac{\pi^2}{6}=\prod_{p} \Big(1-\frac{1}{p^2}\Big)^{-1},$$ the fact that $\pi^2$ is irrational implies that there are infinitely many primes.
A: The fact that π is irrational has few direct applications. However the techniques used to prove this, or rather used to prove the stronger statement that it is transcendental, have many applications. For example, Baker proved that 1 and the logs of algebraic numbers are linearly independent over algebraic numbers except in trivial cases. (This includes the fact that π is irrational as a special case because π = log(-1)/i.) Baker used his theorem to give effective bounds on the solutions of Diophantine equations and to solve Gauss's class number problem for imaginary quadratic fields, among other things. See Baker's book on transcendental number theory for more details. 
A: The fact that one cannot square the circle was proven as a corollary of the fact that pi is transcendental. http://en.wikipedia.org/wiki/Squaring_the_circle
