Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass?

Of course, $\pi$ is not constructible because it is transcendental and so is not a root of *any* polynomial with rational coefficients. But is there a simple direct proof that $\pi$ is not a root of polynomial of degree $2^n$ with rational coefficients?

The kind of proof I seek is one by induction on the height of a tower of quadratic extensions, one that ultimately relies on a proof that $\pi$ is not rational. Does any one know of a proof along these lines or any other direct proof?

I just want a direct proof that $\pi$ is not constructible without appealing to transcendence.