Chapter 4.1, Problem 32E

(a) Let A_n be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle $2\pi /n$, show that

$A_n=\frac{1}{2}n{r}^2\sin \left(\frac{2\pi }{n}\right)$

(b) Show that ${\lim }_{n\to \infty }{A}_n=\pi r^2$. [Hint: Use Equation 2.4.2 on page 145.]