But we, claiming a greater share of wisdom than the bees, will investigate a somewhat wider problem, namely that, of all equilateral and equiangular plane figures having the same perimeter, that which has the greater number of angles is always greater, and the greatest of them all is the circle having its perimeter equal to them. In a sense it is very satisfying to have the circle as a solution to Statement 1. The circle is unique in its roundness. It's symmetric in every straight line passing through its center. The circle is perfect and, thus, is a good candidate to satisfy the Isoperimetric Theorem.
Polygons are less smooth than a circle, less perfect one can say.
Still in every family of polygons there is one which is less imperfect than the rest. Each of the statements above has an equivalent where the area is given. As in Lemma 2, among all plane curves of fixed length with fixed endpoints , a circular arc encloses a maximum area between it and the line joining its endpoints. Oxford University Press, NY, 1996. The Enjoyment of Mathematics , Dover Publications, 1990.