NAME
MATH 111 - Calculus I
Project 1 - Using Derive
Using Limits to Approximate the Circumference and Area of a Circle
A regular polygon of
n
sides is inscribed in a circle of radius
R
.
1. Show that the perimeter of the polygon is given by
P
(
๎
) = 4
๎R
๎
๎
sin
๎
2
where
๎
is the central angle subtended by one side of the polygon. Also, give the angle
๎
in general
as a function of
n
.
2. Use your formula to show that a circle of radius
R
has circumference 2
๎R
.
Hint:
lim
x
!
0
sin
x
x
= 1
3. Modify you approach in the previous two parts to show that a circle of radius
R
has area
๎R
2
. (
Hint:
๎rst de๎ne a function
A
(
๎
) for the area, then take the limit)