Chapter 7.4, Problem 54AYU

Refer to Problem 109 in section 6.5.

1. Show that the area of a regular dodecagon is given by $K=3\cot \left(\frac{\pi }{12}\right)a^2$ or $K=12\tan \left(\frac{\pi }{12}\right)r^2$, where $a$ is the length of one of the sides and $r$ is the radius of the inscribed circle.

2. Given that each interior angle of a regular n-sides polygon $\left(n\ge 3\right)$ measures $\frac{\left(n-2\right)\cdot {180}^{\circ }}{n}$, generalize these formulas for any such polygon.

Problem 109 in section 6.5.

Area of a Dodecagon Part I A regular dodecagon is a polygon with $12$ sides of equal length. See the figure. The area of a regular dodecagon is given by the formula $A=12r^2\tan \frac{\pi }{12}$, where r is the apothem. Find the exact area of a regular dodecagon whose apothem is $10$ inches.