How Many Sides Does a Circle Have?

A joke about the word side

Two is the riddle's answer: the inside and the outside. It works because "side" has more than one meaning. In geometry a side is a straight segment of a polygon, so a triangle has three and a square has four. A circle has no straight segments at all, which means that in the strict sense it has none. The riddle switches to the looser sense of "side" as a face or a surface and counts the two regions that the curve keeps apart.

What mathematicians would say

There is real mathematics behind the joke. A circle is a simple closed curve, and the Jordan curve theorem says that every such curve drawn on a flat surface divides it into exactly two regions, one inside and one outside. So "two" is a true count of the regions, even though it does not count sides.

Listeners who want a strict answer usually say none, because a circle is not a polygon. Others say one, counting the whole curve as a single curved edge, and a few say infinitely many. The riddle accepts none of these as its punch line, but none of them is a mistake in its own terms.

Polygons that creep toward a circle

Adding sides is how Archimedes measured a circle. Around the third century BC he worked with regular polygons of 6, 12, 24, 48 and finally 96 sides and showed that pi lies between 223/71 and 22/7, which is about 3.1408 and 3.1429. Worked out exactly, a 96-sided polygon drawn inside a circle measures about 3.1410 times the circle's width around its edge, and one drawn outside about 3.1427. Pi itself is about 3.1416, so the two polygons trap it closely.