Squaring the circle

Abstract

Constructively transforming a circle into a square is often referred to as squaring the circle. In this case squaring means to find a square with the same area as a given figure. We still refer to the square root of a number, meaning we are looking for the length of the edge of a square with the given number as area.

‘‘Squaring the circle’’ has become more or less a catch line for something which is impossible or unsolvable. Indeed, the problem cannot be solved with compass and straightedge methods. But as with the other problems, it has a rich history. Although we have already mentioned that the quadratrix can be used to find a square with an equal area to a circle, the nature of this problem is completely different.

We will now explain some solutions that were put forward to address the problem. As in the previous chapters, we will encounter solutions to the problem which cannot be accomplished with compass and straightedge methods, but other geometrical constructions are possible.

We now know that the area of a circle is given by $S=\pi r^{2}$. Now suppose that a square with edge a has the same area. Then $a^{2}=\pi r^{2}$, from which $a=\sqrt{\pi}r$. The problem of squaring the circle is thereby reduced to constructing a line segment with length $\sqrt{\pi}$, using compass and straightedge methods. Because $\sqrt{\pi}$ is the geometric mean of π and 1, it suffices to construct a line segment with length π (see also Construction 3).