**Introduction:**
Area is a measure of the amount of

space contained inside a closed figure.

Perimeter is a measure of the distance around a closed figure. In this lesson we will examine these concepts for a circle. In particular, the

perimeter of a

circle is called the

circumference.

**The Lesson:**

To derive a formula for the

area of a circle, we will examine a diagram of a (partial)

regular polygon with sides of

length *s* and number of sides

*n*. A diagram is shown below.

Assume that

point O is the center of the

regular polygon and

*r*, the distance from the center to a vertex, is called the

radius of the polygon. This means

*r* is the

radius of a

circle and O is the center of a

circle into which a

regular polygon can be inscribed. The

perimeter of the

polygon is the sum of the

*n* sides of

length s and is given by

*ns*. As the number of sides in the

polygon increases, the

polygon fits more exactly into the

circle (becomes closer to the actual shape and size of the circle). We will derive a formula for the

area of the

polygon in terms of the

radius *r* and use this to calculate the

area of a circle.

**The Polygon:**

Triangle AOB is a

__central triangle__. There are

*n* such triangles in this polygon, one for each side.

Since there are *n* such triangles in our polygon:

AOB has a measure of

MOB has a measure half as large or

Since AB is a

side of measure

*s*,

MB has a measure of

and

A =

A =

**The Circle:**

Below we show two tables of values for

for two large values of

*n*.

As you can infer from the diagram below, the more sides we use in the polygon, the closer the

polygon will get to matching the

edge of the circle.

Combine this with the fact that the value of

p for larger and larger values of

*n*, we can conclude that our formula for the total

area of the

polygon is

A =

pr

^{2}.

Similarly, we can use

or

to determine the formula for the

circumference of a circle.

The

perimeter of the

polygon is

*ns* where

*n* is the number of sides. It happens that

p as

*n* gets larger allowing us to conclude that our formula for the total

perimeter of the

polygon is

2

pr.

**Summary: **

**
**

We now have the following two formulas:

- The area of a circle is given by A = pr
^{2} where *r* is the length of the radius of the circle.

- The circumference of a circle is given by C = 2pr where
*r* is the length of the radius of the circle. Notice that the circumference could also be written as C = pd where *d* is the length of the diameter since d = 2r.

**Let's Practice:**

- A circle has a diameter of 8 feet. What are the area and circumference of this circle?

Notice that we are given a

__diameter__ of 8 which means that the

radius is 4.

A =

pr

^{2} =

p(4)

^{2} = 16

p square feet

C = 2pr = 2p(4) = 8p feet

It is typical to leave answers in terms of p unless there is a need for a decimal approximation of the answer.

- A right triangle is inscribed in a circle of radius 6 inches as shown in the diagram. What is the area of the shaded region?

The strategy is to find the

area of the

triangle and subtract it from the

area of the circle.

A

_{triangle} = (½)bh = (½)(12)(5) = 30

square inches

A

_{circle} =

pr

^{2} =

p(6)

^{2} = 36

p square inches

We have the

area of the shaded region as 36

p – 30

83.1

square inches.

- A square is inscribed in a circle of circumference 10p. What is the area of the shaded region? A diagram is given below.

The strategy is to subtract the

area of the

square from the

area of the circle.

C = pd

C = 10p

pd = 10p

d = 10

In our diagram, AC is a

diameter of the circle. Therefore, AC = 10.

We will now use this information along with the

Pythagorean Theorem to find the

length of a

side of the square, which we have labeled

*x* in the diagram.

2x

^{2 }= 100

x

^{2} = 50

x

7.07

^{
}

A_{square} = 50

A_{circle} = pr^{2} = p(5)^{2} = 25p

The

area of the shaded region is

25

p – 50

28.5