Chapter 7.3, Problem 50E

Exercises deal with a regular polygon.For $n\ge 3$, a regular n-gon is an n-sided polygon, all of whose sides are equal in length and all of whose angles are equal in measure.Inscribed polygon.A regular n-gon inscribed in a circle of radius r has n vertices that are equally spaced on the circle.The adjoining figure is an inscribed octagon $(n=8)$on a circle of radius r.a.Show that the area A of a regular n-gon inscribed in a circle of radius r is $A=\frac{n{r}^2}{2}\sin \frac{2\pi }{n}$ .

b.What is the area of a disc of radius $r=5$ ?

c.Calculate A from part (a) with $r=5$ and $n=4,10,50$ and $\text{1}00$ .Round your answer to four decimal places.What do you observe?