The apothem of an octagon is a segment from the center of the octagon to the center of one of the sides. Given the length of the apothem of an octagon is 6 feet, construct the octagon.

By looking at the isosceles triangles that compose the octagon, we can determine the lengths of the sides of the octagon and use them to construct the octagon.

We know that the measure of each angle of a regular octagon is 135 degrees. So, the measure of the congruent base angles of the isosceles triangles is 67.5 degrees each and the last angle's measure is 45 degrees. Now, we can look at one of the two congruent right triangles that create the isosceles triangle.

So, the length of the base is tan 67.5 = 6 / x. Since tan 67.5 is approximately 2 .4142, x is almost 2.5 feet. Thus, each side of the octagon is 2 x = 5 feet. The length of the third side of the right triangle is a = 6 / (sin 67.5) which is approximately 6 .5 feet.

Now to construct the octagonal base of a gazebo that is 12 feet across, we have at least two options from the information that we found.

Option I

Choose a center. Place 4 boards of length 12 feet, that intersect at the center and are spaced 45 degrees apart.

At the endpoints of each of these, place the midpoint of a board of length 5 feet. The 5-feet boards should be perpendicular to the 12-feet boards.

Option II

Choose a center. Place 4 boards of length 13 feet, that intersect at the center and are spaced 45 degrees apart.

The ends of the boards are the 8 vertices of the octagonal base of the gazebo.

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