The largest pentagon that can be fitted inside a regular hexagon:

At least 2 adjacent vertices of the pentagon must touch the hexagon

Proof: At least 2 vertices must be touch the hexagon, or else you could grow the pentagon until two did. If those two were not adjacent (eg they were vertices 1 and 3 rather than 1 and 2), you could grow the pentagon until the two vertices were on opposite vertices of the hexagon (across its diameter). But this is impossible as the resulting pentagon is not inside the hexagon - therefore 2 adjacent vertices of the pentagon must touch the hexagon.

The two alternatives now are for these adjacent vertices of the pentagon to lie on two adjacent sides of the hexagon (eg sides 1 and 2), or on two alternate sides (eg sides 1 and 3), including the vertices of those sides. Let's start with them on adjacent sides.

The lower two vertices of the pentagon are on adjacent sides of the hexagon. We have placed the pentagon symmetrically to start with, and grown it so 2 further vertices touch the hexagon. However the top vertex does not touch.

We can therefore make the pentagon a tiny bit bigger by raising it slightly (so the top vertex heads towards the top vertex of the hexagon, but doesn't touch), and then tilting:

This is bigger, because the dotted line is now at an angle between the 2 parallel sides of the hexagon. But the we can still raise the pentagon a bit more and tilt it - in fact we can carry on doing this until the top vertex touches the hexagon side.

Therefore the biggest pentagon has 2 adjacent vertices on alternate sides of the the hexagon. We can now optimise this case.

We show here the pentagon with its lower 2 vertices touching alternate sides of the hexagon:

The pentagon is symmetrically placed. This will be optimum - consider the quadrilateral formed from the lower 4 vertices of the pentagon. Opposite vertices are on parallel sides of the hexagon. Any attempt to tilt the pentagon will reduce its size.

We'll prove later that the top vertex is within the hexagon. Now let's work out the side length of this pentagon, which is within a hexagon of unit side length:

Now to check that the top vertex of the pentagon is within the hexagon:

which is true.

page date: 26Oct05.      I enjoy correspondence stimulated by this site. You can contact me here.