# hexagon diagonal help

• Dec 2nd 2013, 04:15 PM
vegasgunner
hexagon diagonal help
https://data.artofproblemsolving.com...8b3c29a902.png

I have solved the longer diagonal to be 2 if you plug in 1 as the side length of the hexagon, but I do not know how to calculate the shorter diagonal's length
• Dec 2nd 2013, 04:51 PM
HallsofIvy
Re: hexagon diagonal help
If you draw the 6 "long diagonals" you divide the hexagon into 6 equilateral triangles. That is why the length of the "long diagonal" is 2- it is made of two of those sides of length 1. The "short diagonal" is made of two altitudes of equilateral triangles. You can find the length of an altitude of an equilateral triangle by observing that it divides the equilateral triangle into two right triangles have a side of length 1 as hypotenuse and half a side, of length 1/2, as a leg. Use the Pythagorean theorem to find the length of the other leg, the altitude.
• Dec 2nd 2013, 05:58 PM
vegasgunner
Re: hexagon diagonal help
Quote:

Originally Posted by HallsofIvy
If you draw the 6 "long diagonals" you divide the hexagon into 6 equilateral triangles. That is why the length of the "long diagonal" is 2- it is made of two of those sides of length 1. The "short diagonal" is made of two altitudes of equilateral triangles. You can find the length of an altitude of an equilateral triangle by observing that it divides the equilateral triangle into two right triangles have a side of length 1 as hypotenuse and half a side, of length 1/2, as a leg. Use the Pythagorean theorem to find the length of the other leg, the altitude.

thanks a lot hallsofivy :) I have solved it to be sqrt 3/2