Which results depend on the irrationality of $\pi$?

Question

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it funny too (I just hope she never gets to see this question).

Of course the first thing I noticed was the gross mistake at 9 o-clock. But then this mistake got me thinking about what would happen if $\pi$ were a rational number. By this I mean,

What sort of important results depend crucially on the irrationality of $\pi$?

The only example I was able to come up with is the good old greek problem of the squaring of the circle, which basically asks for the constructibility of $\sqrt{\pi}$ and thus if $\pi$ were rational, its square root would be constructible and thus the problem would not be impossible.

NOTE

I edited the question title and a part of my question because as was pointed out in some of the comments, that part didn't make much sense. Although I didn't know that at the moment, so it wasn't such a bad thing that I included that "nonsense" in my question at first. But as Bruno's answer explains, some part of my original misunderstanding can be given some sense after all.

Answer 1

If that happens, the circle would not be a differentiable manifold ... And so most manifolds would not be smooth ... The world would be very painful to live.

Answer 2

Edit: the question was changed by the time I had finished writing this post, but I'll leave it up.

I'm going to say that this question, as I interpret it, does not really make much sense. At the very least, it is not a mathematical question. From my understanding, it can be interpreted in two different ways (which I am wording very loosely):

1. Could $\pi$ have a different value?
2. In a different Universe, could $\pi$ have a different value?

Question 1, which I think is the one you mean to ask, has a simple answer: mathematics would be inconsistent, as pointed out by Brian in the comments. The simple reason is that $\pi$ can be proven to be irrational (and in fact transcendental), hence if we could also prove it to be rational, we'd be in big trouble.

Question 2 is more interesting but requires a little interpretation. We can define $\pi$ to be the circumference of a circle of diameter $1$ in the Euclidean plane. We can mimick this definition by changing the rules of the game a little bit. For instance, if we change the metric on the plane, then a "circle" takes on a whole new meaning. For instance, in the taxicab metric, a circle of diameter $1$ is simply a square with side length $1$, and its circumference is $4$. Thus in a taxicab Universe, $\pi$ would have a different (and rational!) value, but that's simply because it would have a different definition...

There are also spaces in which the circumference of a circle of fixed radius varies with the position of the circle in the space.