The missing length is the square root of π
A circle with radius r has area πr². To make a square with the same area, its side must be √π times the circle’s radius: squaring that length gives the required area. So the puzzle is not simply about drawing a square; it is about whether the allowed tools can construct that particular length exactly.
What compass and straightedge can produce
In classical construction, a straightedge draws lines and a compass draws circles. New points come from intersections of those lines and circles. Finding an intersection amounts to solving equations that are linear or quadratic, so each construction step can introduce only the kinds of lengths built from rational numbers through arithmetic and square roots. Such lengths are called algebraic numbers.
π is different: it is transcendental, meaning it is not a solution to any polynomial equation with whole-number coefficients. Its square root is also transcendental; if √π were algebraic, squaring it would make π algebraic too. Thus the side needed for the equal-area square lies outside the lengths these constructions can produce.
Approximation is a different task
The limit is exactness, not accuracy. A constructed square can come remarkably close in area, and practical methods can make the discrepancy too small to notice. But no matter how many permitted steps are used, they cannot produce the precise side length √π times the radius. The circle can be squared only by relaxing the rules—for example, by allowing a different tool or accepting an approximation.
