Chapter 7.6, Problem 101AYU

Area of an Octagon

(a) The area $A$ of a regular octagon is given by the formula $A=8r^2\tan \frac{\pi }{8},$ where $r$ is the apothem, which is a line segment from the center of the octagon perpendicular to a side. See the figure. Find the exact area of a regular octagon whose apothem is 12 inches.

(b) The area $A$ of a regular octagon is also given by the formula $A=2a^2\cot \frac{\pi }{8}$, where $a$ is the length of a side. Find the exact area of a regular octagon whose side is 9 centimeters.