high school geometry
the shorter diagonal of a regular hexagon is 6. What is the length of the longer diagonal?
In order to answer this clearly, I'm going to need you to look at this image http://i.imgur.com/nJh8x.png
Now, you know that the interior angles of a regular polygon must add up to 180(n-2), which in this case, with n=6, gives you 720. That means that each of the interior angles is 120 degrees. You can see in the image that I marked angle B as being 120 degrees.
Now, triangle ABC is an isosceles triangle, because two of its sides are the sides of the hexagon. That means that the other two angles are equal. Since the angles of a triangle must add up to 180 degrees, that means that each of the other two are 30 degrees, like it's written in the figure.
Now look at angle C. As you can see, part of it, angle ACB, is 30 degrees, like we just explained. We also know that the whole thing must be 120 degrees. That means that what's left, angle ACD, must be a right angle, or 90 degrees. So triangle ACD is a right triangle, and you can use what you know about right triangles in it.
Now let's look at what type of right triangle it is. Look at angle CDA. It's formed by the long diagonal of the hexagon, which cuts angle D exactly in half. That means that angle CDA is exactly half of 120 degrees, so it is 60 degrees. Since all angles in a triangle must add up to 180, we know that the last angles, angle DAC must be 30 degrees. So this means that triangle ACD is a 30-60-90 triangle.
You know how the sizes of the sides relate to each other in a 30-60-90 triangle, and you know that side AC is 6, so now you should be able to calculate the size of side AD, which is the long diagonal of the hexagon.