Largest small polygon: max area at fixed diameter | Tangente

For a given number n of sides, this gives the largest small n-gon.

For a quadrilateral, the area is e × f × sin(θ)/2, where e and f are the lengths of the diagonals and θ is the angle between them (see page 18).

The maximum occurs when e = f = sin(θ) = 1.

This corresponds to infinitely many quadrilaterals, including the square.

For odd n, Karl Reinhardt proved in 1922 that the unique solution was the regular polygon.

For n = 6, the unique solution is not regular. Ronald Graham showed in 1975 that it consists of an irregular equidiagonal pentagon with an isosceles triangle attached to one of its sides.

Its area, approximately 0.67498, is a root of a tenth-degree equation! He conjectured that the same construction worked for every even number n of sides: an isosceles triangle attached to an equidiagonal (n – 1)-gon. This was proved in 2007 by Tamás Szabo, a Hungarian mathematician.