From: Emilio

Hi, I need to find the measure of the sides of the hexagon that the area is 58 square units.

Hi Emilio.

I'm going to assume that your question is about a regular hexagon (a hexagon where all the sides and the angles are the same). Here's a diagram to help us:

You should be able to figure out the angles α and β on your own. What your question is asking for is the length of a side of the hexagon, which is 2b in our diagram.

Can you see from symmetry that there have to be eleven other triangles in the hexagon that are congruent to the one I've drawn? That's because the segment a is what is called the apothem - it is a perpendicular bisector of the side and ends at the center.

The area of the triangle I've drawn is ½ab, of course, so the total area of the hexagon is twelve times that: 6ab.

If we say that the total area is A, then A = 6ab.

We need to solve for 2b, so that would be 2b = A / 3a. But what is a?

Since this is a right-triangle, let's use the angle β to help us: tan β = a / b, so a = b tan β.

If we substitute that into our earlier expression, 2b = A / 3(b tan β).

Since you are given A (58 square units) and can easily determine β for this shape, you can solve for b now.

Hope this helps,
Stephen La Rocque. >