Reuleaux Shapes and Why Manhole Covers Are Round

The classic interview riddle "why are manhole covers round?" has a real engineering answer — a circle can never fall through a hole cut to its own diameter, because no matter how you tilt it, its width in every direction is the same. What trips people up is the follow-up: is a circle the only shape with that property? It isn't. A whole family of curved shapes, called curves of constant width, share it, and the simplest one is the Reuleaux triangle.

Build one by taking an equilateral triangle and replacing each straight side with a circular arc, centred at the opposite corner, with radius equal to the triangle's side length. The result looks like a rounded triangle, but measure its width in any direction — the distance between two parallel lines squeezing it from opposite sides — and it comes out equal to the original side length, every time. That is because each arc is centred on the corner across from it, so the "far point" you hit rolling across the shape is always exactly one side-length away.

This matters beyond trivia: constant-width shapes can be used as rollers (a Reuleaux triangle rolls a flat plate smoothly, even though its centre bobs up and down slightly, unlike a circle's), and the same idea underlies drill bits that cut nearly-square holes. In interviews, the manhole question is really testing whether you reach for "circle" as the only answer or recognize that the property being asked about — constant width — is what actually prevents the cover from falling, and that other shapes satisfy it too.