Chapter 4.5, Problem 62E

Circumscribed polygon. A regular circumscribed n -gon about a circle of radius r has n vertices and its sides are tangent to n equally spaced points on the circumference of the circle. The adjoining figure is a circumscribed hexagon

$\left(n=6\right)$ about a circle of radius r. Use this figure.

a. Show that the area A of a regular n -gon circumscribed about a circle of radius r is:

$A=n{r}^2\tan \frac{\pi }{n}$.

b. What is the area of disc of radius $r=10$?

c. Calculate A by using the formula in part with $r=10$ and $n=4,10,50,and100$. Round your answer to four decimal places. What do you observe?