What is the ratio of the measure of a regular hexagon's longest diagonal to the measure of its perimeter? Express your answer as a common fr

The answer is 1/6.

Split the hexagon into 6 equilateral triangles, and let the sides of the hexagon be s.

The longest diagonal would be the one between two opposite vertices, which is comprised of the sides of 2 equilateral triangles with side length s, so its length is 2s.

The perimeter of the hexagon is 6 times the side of the hexagon or 6s. 

So, the ratio is \({2s \over 6s} = {\color{brown}\boxed{1 \over 3}}\)