I am assuming this is a regular octagon. If you figure out the measure of all of its 8 interior angles, you divide 360 by 8 to get that each triangle has a vertex angle of 45. But we want half of that triangle so we can find the apothem measure. Therefore, the vertex angle is 22.5. We have the fact that this "new" half triangle is a right triangle, therefore, that 3rd angle measure 67.5. Because we have half of the triangle, the base measure is half of 5 which is 2.5. Now we can use the tan67.5 = a/2.5 to find a, the apothem. We find that that is 6.0. Now we have the apothem, now we need the perimeter which is 40. The area formula for the regular polygon is A = 1/2ap, so ours looks like this: [tex]A= \frac{1}{2}(6)(40) [/tex] and the area is 120 units squared.
