Maximum Polygon Area

A polygon has sides of fixed length, but is flexible about its vertices.

How should it be arranged for maximum area?

For maximum area, all vertices should lie on a circle. The polygon is then called a "cyclic polygon". It does not matter in which order the polygon sides occur.

First proof (from David Eppstein): arrange the vertices so they lie on a circle, and 'glue' the arc-shaped (red) caps onto each side of the polygon:

We can still change the shape of the polygon, with the (red) caps on each of its sides, as shown below. The total outside perimeter (round the outside of all the red bits) does not change.

The area within the outside perimeter comprises the area of the polygon plus the area of all the red caps. The area of the red caps does not change as we alter the polygon. Therefore if we maximise the area within the outside perimeter we will maximise the area of the polygon.

The maximum area within the outside perimeter occurs when the outside perimeter is a circle. So the maximum area for the polygon occurs when the outside perimeter is a circle. So the polygon vertices lie on a circle.

Reference from http://www.drking.org.uk/hexagons/misc/polymax.html