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jmc

geometry intermediate

Problem

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 fraction.
Solution
We draw a regular hexagon and one of its longest diagonals: Now we draw in the other two long diagonals. We have split the hexagon into six equilateral triangles, which are congruent by symmetry. We see that two side lengths make up one long diagonal. Let the hexagon's side length be , then, its long diagonal has length and its perimeter has length . The ratio of its long diagonal to its perimeter is hence .
Final answer
\frac{1}{3}