Solution to the Problem: The answers are:
The width of each panel is 32 inches and
the area of the octagon is 34.3698 square feet.
(The advertisement never mentioned the length of each panel --which is why my daughter sent the "problem" to me, but the ad gave 34.4 square feet as the area).
To solve the problem, I drew the three diagonals of the hexagon which intersect in the center, thus forming six equilateral triangles.
Letting x represent the side of the panel (and therefore, the length of each side of the six equilateral triangles), you can set up the following formula for the area of the hexagon:
To solve for the area of the octagon, I again drew the four diagonals
which intersect in the center of the octagon. This forms 8 isosceles
triangles with angle measurements 67.5, 67.5, and 45 degrees. The side
opposite the 45 degree angle is 2.668 feet (the length of the panel).
Draw an altitude from the center of the octagon to one of the
eight panels. This divides the triangle into two smaller right
triangles whose acute angles measure 22.5 and 67.5 degrees. The leg
opposite the 22.5 degree angle is 1.334 (half the length of the panel).
Call the other leg h.
Use the tangent function to solve for h:
