Squaring the Circle by Compass and Straightedge Construction is Impossible

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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.

$\blacksquare$


Historical Note

The crucial final step in the proof of the age-old classic problem of Squaring the Circle was made by Ferdinand von Lindemann, who proved in $1882$ that $\pi$ is transcendental.


Sources