There is a very long baby gate, which is advertised as being able to be used as a play yard for children ("A gate that doesn't quit"). This 6-panel play yard "provides 18-1/2 square feet of protected play area when it is set up as a hexagon." The ad continues by saying that the play area expands to so many square feet simply by adding a 2-panel expansion kit.

Assuming that all the panels are the same width, determine the width of each panel. Also, determine how many square feet of space would be provided by the octagon (by adding the 2-panel expansion kit to the hexagon).


Solution:
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:



Correctly solved by:

1. Keith Mealy Cincinnati, Ohio
2. Chip Crawford College of William & Mary
3. Walt Arrison Philadelphia, Pennsylvania
4. Diana Xing Fort Collins, Colorado
5. John C. Funk Ventura, California
6. Bob Hearn Winchester, Virginia
7. Richard K. Johnson La Jolla, California
8. Tony Wu Fort Collins, Colorado
9. Brandon Copple Winchester, Virginia
10. Renata Sommerville Austin, Texas