Most Area, Least Fence

Why the circle is special

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Isoperimetric inequality: $4\pi A \le L^2$, with equality only for the circle.
• Regular polygons approach the circle’s area as the number of sides grows.
• Against a straight coast, the best shape is a semicircle (Dido’s problem).

Steps

1. A square pen
2. More sides
3. The isoperimetric inequality
4. A round pen
5. Dido’s problem
6. Dido’s kingdom