---
tag: 0X7CVDE3BG
kind: statement
role: theorem
name: Impossibility of squaring the circle
msc: [12F05]
uses: [0BSXRER81B, 0G0F7FGCGP, 0SSSM1D475]
sources:
  - url: https://en.wikipedia.org/wiki/Squaring_the_circle
    locator: Impossibility
---

## Statement

The number $\sqrt{\pi}$ is not a [[constructible number|0FMN2MN7TM]].
So the circle cannot be [[squared|0BSXRER81B]] with straightedge and compass.

## Remarks

The proof: every constructible number is [[algebraic over $\mathbb{Q}$|0SSSM1D475]]. If $\sqrt{\pi}$ were constructible, then $\pi = (\sqrt{\pi})^2$ would be constructible and so algebraic.
This contradicts the [[transcendence of $\pi$|0G0F7FGCGP]].

## History

The impossibility was proved in 1882, as a consequence of Lindemann's proof that $\pi$ is transcendental.
It had been known for decades that the transcendence of $\pi$ would imply the impossibility.

## Explanations

### L1 human

A square with the area of the unit circle has side $\sqrt{\pi}$.
Constructible numbers are algebraic, and $\pi$ is not, so this side cannot be constructed.