# Given a fixed perimeter, which shape will have the maximum area?

I think the answer is a circle. If so, then what is the rigorous prove?

• It is correct :D – Eli Elizirov Aug 7 '13 at 12:16

## 3 Answers  I have attached images with equations in word format

You are correct. It follows from the Isoperimetric Inequality.

A circle gives the maximum area for a given perimeter.

So the triangle that gives the maximum area for a given perimeter is an equilateral triangle. The quadrilateral that gives the maximum area for a given perimeter is a square. The pentagon that gives the maximum area for a given perimeter is a regular pentagon. The n-gon that gives the maximum area for a given perimeter is a regular n-gon.

Additionally, as the number of sides, n, grows in a regular n-gon, the area will increase for a given perimeter. Thus of a square and an equilateral triangle with the same perimeter, the square will have more area. But of a square and a regular pentagon with the same perimeter, the pentagon will have the larger area.

So the most efficient shape for any perimeter is a circle. This is because you could imagine a circle as a regular n-gon with an infinite number of sides.