Two different types of hexagons exist -- regular, with identical sides and angles, and irregular. Both types of hexagons have nine diagonals. Most, if not all, diagonals of an irregular hexagon are likely to be of different lengths; like all irregular polygons, an irregular hexagon does not allow for general calculations of its diagonals unless you are given all sides and all angles. As long as you know the length of a side, on the other hand, you can easily calculate the length of the diagonals of a regular hexagon.
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Consider a regular hexagon with vertices ABCDEF and known side length X. The hexagon has three identical long diagonals, connecting diametrically opposite vertices that are separated by two other vertices, and six identical short ones, connecting vertices that are separated by only one other vertex. Call the long diagonal between A and D "Y" and the shorter diagonal between A and E "Z."
Multiply the side length by two to get the long diagonal:
Y = 2X
This is because a regular hexagon is composed of six equilateral triangles, which means the radius of the circle it can be inscribed in is identical to its side; the long diagonal of the hexagon corresponds to the circle's diameter, which is double its radius.
Consider the triangle AEF, formed by the short diagonal and the two sides adjacent to it. Because this is an isosceles triangle and the angle between the two identical sides is 120 degrees, you can determine that the other two angles of this triangle are both 30 degrees.
Consider the triangle AED, formed by the short diagonal, the long diagonal and the side adjacent to it. This triangle's AED angle and the previous triangle's FEA angle, together, add up to the hexagon's FED angle; since FEA is known to be 30 degrees and FED to be 120 degrees, AED is 90 degrees. This makes the AED triangle right-angled with two known sides, X and Y.
Apply Pythagoras' theorem to the AED triangle to find the short diagonal:
Z = sqrt( Y^2 - X^2 )
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