You have a recipe for an 8-in pizza but you want to make a pizza for your 16-in round pan. Will doubling your recipe be enough to fill the larger pan?

In this concept, you will learn to calculate the area of circles and the area of sectors of circles.

### Area

**Area** is the amount of two-dimensional space a figure takes up. In other words, area is the space contained within a circle’s circumference. The formula for the area of a circle is \begin{align*}A = \pi r^2\end{align*}

A **sector** is a part of a circle with radii for two sides and part of the curved circumference as another. Sectors look like pie slices. Look at the diagram below. The shaded area is the sector of the circle.

For now, the area of a sector can be found by finding the proportion of the circle. In the diagram above, the blue sector is \begin{align*}\frac{1}{4}\end{align*}

Let’s look at an example.

What is the area of the circle below?

First, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi(12)^2
\end{array}\end{align*}

Next, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi(12)^2\\
A &=& \pi(144)\\
A &=& 452.4
\end{array}\end{align*}

The answer is 452.4.

The area of the circle is 452.4 cm^{2}.

Let’s look at another example.

The area of a circle is 113.04 square inches. What is its radius?

First, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
113.04 &=& \pi r^2
\end{array}\end{align*}

Next, divide both sides by

.\begin{align*}\begin{array}{rcl}
113.04 &=& \pi r^2\\
\frac{113.04}{\pi} &=& \frac{\pi r^2}{\pi}\\
r^2 &=& 35.98
\end{array}\end{align*}

Then, take the square root of both sides to solve for \begin{align*}r\end{align*}

\begin{align*}\begin{array}{rcl}
r^2 &=& 35.98\\
r &=& \sqrt{35.98}\\
r &=& 6
\end{array}\end{align*}

The answer is 6.

The radius of the circle is 6 in.

Let’s look at an example of finding the area of a sector.

What is the area of the figure below?

First, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi(8.5)^2
\end{array}\end{align*}

Next, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi (8.5)^2\\
A &=& \pi (72.25)\\
A &=& 226.98
\end{array}\end{align*}

Then, knowing the figure is one-quarter of a circle, divide this area by 4.

\begin{align*}\begin{array}{rcl}
A_{\text{sector}} &=& \frac{226.98}{4}\\
A_{\text{sector}} &=& 56.7
\end{array}\end{align*}

The answer is 56.7.

The area of the sector is 56.7 in^{2}.

### Examples

#### Example 1

Earlier, you were given a problem about the expanding pizza.

You want to know if you simply need to double the ingredients for your pizza if you want to use a 16-in round pan rather than an 8-in round pan.

First, find the radius of each pizza pan.

\begin{align*}\begin{array}{rcl}
&& 8 - \text{in round pan} \qquad 16 - \text{in round pan}\\
&& r = \frac{d}{2} \qquad \ \ \ \qquad \qquad r = \frac{d}{2}\\
&& r = \frac{8}{2} \qquad \ \ \ \qquad \qquad r = \frac{16}{2}\\
&& r = 4 \qquad \ \ \ \ \qquad \qquad r = 8
\end{array}\end{align*}

Next, find the area of each pizza pan.

\begin{align*}\begin{array}{rcl}
&& 8 - \text{in round pan} \qquad 16 - \text{in round pan}\\
&& A =\pi r^2 \qquad \ \ \ \quad \qquad A = \pi r^2\\
&& A = \pi (4)^2 \quad \quad \ \ \ \qquad A = \pi (8)^2\\
&& A = \pi \times 16 \quad \quad \ \qquad A = \pi \times 64\\
&& A = 50.3 \qquad \ \quad \qquad A = 201.1
\end{array}\end{align*}

Then, divide the area of the 16-in round pan by the area of the 8-in round pan.

\begin{align*}\begin{array}{rcl}
\frac{A_{16-\text{in round pan}}}{A_{8-\text{in round pan}}} &=& \frac{201.1}{50.3}\\
\frac{A_{16-\text{in round pan}}}{A_{8-\text{in round pan}}} &=& 4
\end{array}\end{align*}

The answer is 4.

The 16-in pizza pan is four times the size of the 8-in pizza pan, so doubling the ingredients is not enough.

#### Example 2

What is the area of a circle with a diameter of 45 centimeters?

First, find the radius of the circle. Remember the radius is half the diameter.

\begin{align*}\begin{array}{rcl}
r &=& \frac{d}{2}\\
r &=& \frac{45}{2}\\
r &=& 22.5
\end{array}\end{align*}

Next, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi (22.5)^2
\end{array}\end{align*}

Then, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi (22.5)^2\\
A &=& \pi (506.25)\\
A &=& 1590.4
\end{array}\end{align*}

The answer is 1590.4.

The area of the circle is 1590 cm^{2}.

Find the area of each circle by using the given dimension.

#### Example 3

Find the area of each circle with a radius of 2.5 inches.

First, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi (2.5)^2
\end{array}\end{align*}

Next, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi (2.5)^2\\
A &=& \pi (6.25)\\
A &=& 19.6
\end{array}\end{align*}

The answer is 19.6.

The area of the circle is 19.6 in^{2}.

#### Example 4

Find the area of each circle with a radius of 3 feet.

First, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi (3)^2
\end{array}\end{align*}

Next, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi (3)^2\\
A &=& \pi (9)\\
A &=& 28.3
\end{array}\end{align*}

The answer is 28.3.

The area of the circle is 28.3 ft^{2}.

#### Example 5

Find the area of each circle with a diameter of 8 inches.

First, find the radius of the circle. Remember, the radius is half the diameter.

\begin{align*}\begin{array}{rcl}
r &=& \frac{d}{2}\\
r &=& \frac{8}{2}\\
r &=& 4
\end{array}\end{align*}

Next, substitute what you know into the formula for the area of a circle.

\begin{align*}\begin{array}{rcl}
A &=& \pi r^2\\
A &=& \pi (4)^2
\end{array}\end{align*}

Then, solve for the area.

\begin{align*}\begin{array}{rcl}
A &=& \pi (4)^2\\
A &=& \pi (16)\\
A &=& 50.3
\end{array}\end{align*}

The answer is 50.3.

The area of the circle is 50.3 in^{2}.

### Review

Find the area of each circle given the radius.

1. \begin{align*}r =4 \ in\end{align*}

2.

3.

4.

5. \begin{align*}r = 3.5 \ in\end{align*}

6.

7.

8.

9. \begin{align*}r = 7 \ ft\end{align*}

10.

Find the area of each sector given the radius and the angle measure. You may round to the nearest hundredth as needed.

11. 45° angle with a radius of 3 in.

12. 55° angle with a radius of 4 mm

13. 60° angle with a radius of 5 cm

14. 43° angle with a radius of 6 in

15. 70° angle with a radius of 2 in.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.4.