Explanation of Official Guide/Big Book Math Question

[Robert.Bregman]

I am struggling with Question #29 on page 245; the one involving the octagon. Even with the correct answer, I still can't figure out how to solve it.

Please help.

Many thanks,

Robert Bregman

[tommywallach]

Hey Robert,

One of the keys to this question is the word "equiangular" polygon. Because you know the angles are equal, you know the whole thing is symmetrical.

Imagine surrounding the whole thing in a square. You will have now created 4 right triangles (one on each corner). These triangles will have a hypotenuse of root 2, and equal angles in the two angles that aren't 90 degrees (because the original octagon was equiangular). This means they are 45-45-90 triangles.

If you remember your rules, the sides of a 45-45-90 triangle exist in a predetermined ratio: Side-Side-Side*root 2, where the Side*root 2 = the hypotenuse. That means the perpendicular sides of these triangles must be 1.

If you look at the whole square now, it is 3x3, with an area of 9. We also have 4 triangles, each of which has an area = 1/2 * b * h = 1/2 * 1 * 1 = 1/2. Because there are 4 of them, we do 1/2 * 4 = 2. If we take the full 9 of the square minus the 2 of the triangles, we get 7.

The answer is A.

Hope that helps!

-t