Can you draw a square with the exact area of a circle, using only a compass?
Ancient geometers spent over two thousand years trying to construct, using only an unmarked straightedge and compass, a square with exactly the same area as a given circle. Countless attempts failed. Was it simply a matter of not being clever enough, or is the task fundamentally impossible?
Reveal the answer
It's impossible, and the proof took until 1882. Ferdinand von Lindemann showed that pi is a 'transcendental' number — not the solution to any polynomial equation with rational coefficients — and only non-transcendental (algebraic) lengths can ever be constructed with compass and straightedge. Since squaring the circle requires constructing a length involving the square root of pi, the ancient problem was mathematically doomed from the start.
— Ferdinand von Lindemann, Squaring the circle — Proof of pi's transcendence, 1882