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.

$\blacksquare$

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

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks