Squaring the circle

Problem

Given a circle, construct with straightedge and compass alone a square with the same area.

Remarks

For a circle of radius $1$, the square has side $\sqrt{\pi}$. So the problem asks whether $\sqrt{\pi}$ is a constructible number. The answer is no: squaring the circle is impossible.

History

Anaxagoras is the first known Greek to study the problem. Hippocrates of Chios squared some figures bounded by circular arcs, the lunes of Hippocrates. In 1882, Ferdinand von Lindemann proved that $\pi$ is transcendental, and so that the construction is impossible.

Explanations

L1 human

The problem asks for a square with the same area as a given circle, with only an unmarked ruler and a compass. The phrase "squaring the circle" now means to attempt something impossible.