It is famously impossible to square the circle. That is, given a circle, it is impossible, using only a compass and straightedge, to construct a square having the same area as the circle.
I will let you read elsewhere about the exact rules behind compass and straightedge constructions. The punchline is that if you begin with two points 1 unit apart, then you can construct a line segment segment of length if and only if can be created from the integers using the operations , , , and , and by taking square roots.
A circle with radius 1 has area . A square having this same area would have side-length . Because is transcendental, this is not a constructible length. Thus it is impossible to square the circle.
Even though the circle is not squarable, some regions with curved boundaries are. For example, in the fifth century BCE Hippocrates of Chios (no, not that Hippocrates) showed that several lunes are squarable.
Let’s give a brief proof that the shaded lune shown below is squarable. Suppose the large circle has radius 1. Then triangle has area and sector has area . Thus segment has area . The smaller circle has radius . So the semicircle has area . It follows that the lune has area . A square with the same area as the lune would have side-length , which is constructible. Thus the lune is squarable.
It turns out Leonardo da Vinci played around with squarable figures, and he discovered many beautiful examples. Below I’ve included one of Leonardo’s figures (the on the left). I’ve included the center and right-hand figures to give more information on how Leonardo’s design is created.
So here’s the puzzle: show that Leonardo’s figure is squarable.
(Hint: assume that the radius of the large circle is 1. Then find the total area of the shaded regions. Show that the area, and hence the square root of the area, is a constructible number.)