Four small circles equal one big one: the geometry that surprises

Four small circles equal one big one: the geometry that surprises

A basic geometry problem has been circulating on social media and economic forums, raising doubts even among adults. The question: compare the total area of the grey zone in two figures — a square with one large inscribed circle, and the same square with four small inscribed circles. The answer, for anyone who hasn't worked it out, is that both areas are equal.

The explanation is simple: the area of a circle is proportional to the square of its radius. If the large circle has radius R, its area is πR². If the four small circles have half the radius (R/2), each one has an area of π(R/2)² = πR²/4. Four of them add up to 4 × πR²/4 = πR², exactly the same. There's no trick: the intuition that four small objects should take up less space is wrong when the radius is halved, because area depends on the square.

It is striking that a primary-school-level problem sparks debate even on an economics forum. Some comments joke about using artificial intelligence to solve it, while others point out that it was enough to notice that the percentage of grey area over the square is the same in both figures, regardless of the size of the circles. The sarracena: visual perception deceives, but mathematics doesn't lie. Why is it so hard to trust the numbers?

The question that arises is whether this kind of confusion carries over into the real economy, where intuition about percentages, compound interest or scales can also lead to serious mistakes. Perhaps that's why some people argue for strengthening maths education from an early age, before AI does for us what we should know how to do ourselves.