The largest hexagon that can be fitted inside a circle:

Here we are considering a hexagon that may be irregular. By 'the largest hexagon', we mean the hexagon with the largest area. All vertices of the hexagon must touch the circle, because if one did not, we could simply move it out onto the circle to increase the size.

We consider any three adjacent vertices of the hexagon

The area of the triangle shown contributes to the area of the hexagon. The area will be largest when h is largest - which is when the middle vertex is exactly halfway between the other two.

This applies to any 3 adjacent vertices, which means the hexagon must be regular.

page date: 28Oct05.      I enjoy correspondence stimulated by this site. You can contact me here.