Squaring the Circle by Compass and Straightedge Construction is Impossible

Theorem
There is no compass and straightedge construction to allow a square to be constructed whose area is equal to that of a given circle.

Proof
Squaring the Circle consists of constructing a line segment of length $\sqrt \pi$ of another.

From Constructible Length with Compass and Straightedge, any such line segment has a length which is an algebraic number of degree $2$.

But $\pi$ is transcendental.

Hence $\pi$ and therefore $\sqrt \pi$ is not such an algebraic number.

Therefore any attempt at such a construction will fail.