The Largest Small Hexagon

Given any n-gon of unit diameter in the plane, it is natural to inquire how large an area it can have. In 1922, Reinhardt [8] showed that for odd II, the obvious configuration is optimal, namely, among all n-gons of diameter 1, the regular n-gon has the maximum area. It was noted, however, that this is not the case for n even.l In particular, H. Lenz [7] raised the question of finding the largest area, denoted by F6, a plane hexagon of unit diameter can have.2 In this note we answer this question. It is shown that F6 satisfies a 10th degree irreducible polynomial over the integers and is approximately equal to 0.674981. Furthermore, there is a unique unit diameter hexagon with area F6 .