# 9.17: Area of a Circle

**Basic**Created by: CK-12

**Practice**Area of a Circle

Do you remember Miguel? He had just finished working on figuring out the circumference of three different on deck pads for the pitchers to use while they warm up. Let’s look at his dilemma again before we look at the area of the pads.

Miguel’s latest task is to measure some different “on deck” pads for the pitchers to practice with. An on deck pad is a circular pad that is made up of a sponge and some fake grass. Pitcher’s practice their warm-ups by standing on them. They work on stretching and get ready to “pitch” the ball prior to their turn on the mound.

Miguel has three different on deck pads that he is working with. The coach has asked him to measure each one and find the circumference and the area of each.

Miguel knows that the circumference is the distance around the edge of the circle. He decides to start with figuring out the circumference of each circle.

He measures the distance across each one.

The first one measures 4 ft. across.

The second measures 5 ft. across.

The third one measures 6 ft across.

Miguel has completed the first part of this assignment. He knows the circumference of each pad. Now he has to figure out the area of each. Miguel isn’t sure how to do this. He can’t remember how to find the area of a circle. Miguel needs some help.

**This is where you come in use this Concept to help you learn how to find the area of a circle. When finished, we’ll come back to this problem and you can help Miguel figure out the area of each on deck pad.**

### Guidance

You have learned how to calculate the circumference of a circle. Let’s take a few minutes to review the terms associated with circles.

**A** *circle***is a set of connected points equidistant from a center point. The** *diameter***is the distance across the center of the circle and the** *radius***is the distance from the center of the circle to the edge.**

**We also know that the number** *pi, \begin{align*}\pi\end{align*} π,*

**is the ratio of the diameter to the circumference. We use 3.14 to represent pi in operations.**

What does all of this have to do with area?

Well, to find the ** area** of a figure, we need to figure out

**the measurement of the space contained inside a two-dimensional figure.**This is the measurement area. This is also the measurement inside a circle. You learned how to find the circumference of a circle, now let’s look at using these parts to find the area of the circle.

**How do we find the area of a circle?**

The area of a circle is found by taking the measurement of the radius, squaring it and multiplying it by pi.

Here is the formula.

\begin{align*}A= \pi r^2 \end{align*}

*Write this formula down in your notebook.*

We already know that the symbol \begin{align*}\pi\end{align*}

Let’s try it out.

What is the area of the circle below?

**We know that the radius of the circle is 12 centimeters. We put this number into the formula and solve for \begin{align*}A\end{align*} A.**

\begin{align*}A & = \pi r^2\\
A & = \pi (12)^2\\
A & = 144 \ \pi\\
A & = 452.16 \ cm^2\end{align*}

*Remember that squaring a number is the same as multiplying it by itself.*

**The area of a circle with a radius of 12 centimeters is 452.16 square centimeters.**

Some students have formed a circle to play dodge ball. The radius of the circle is 21 feet. What is the area of their dodge ball circle?

**The dodge ball court forms a circle, so we can use the formula to find its area. We know that the radius of the circle is 21 feet, so let’s put this into the formula and solve for area, \begin{align*}A\end{align*} A.**

\begin{align*}A & = \pi r^2\\
A & = \pi (21)^2\\
A & = 441 \ \pi\\
A & = 1,384.74 \ ft^2\end{align*}

**Notice that a circle with a larger radius of 21 feet has a much larger area: 1,384.74 square feet.**

**Sometimes, you will be given a problem with the diameter and not the radius. When this happens, you can divide the measurement of the diameter by two and then use the formula.**

Find the area of each circle.

#### Example A

Radius = 9 inches

**Solution:\begin{align*}254.34\end{align*} 254.34 sq. in.**

#### Example B

**Radius = 11 inches**

**Solution: \begin{align*}379.94\end{align*} 379.94 sq. in**

#### Example C

Diameter = 8 ft.

**Solution: \begin{align*}50.24\end{align*} 50.24 sq. feet**

Here is the original problem once again.

Do you remember Miguel? He had just finished working on figuring out the circumference of three different on deck pads for the pitchers to use while they warm up. Let’s look at his dilemma again before we look at the area of the pads.

Miguel’s latest task is to measure some different “on deck” pads for the pitchers to practice with. An on deck pad is a circular pad that is made up of a sponge and some fake grass. Pitcher’s practice their warm-ups by standing on them. They work on stretching and get ready to “pitch” the ball prior to their turn on the mound.

Miguel has three different on deck pads that he is working with. The coach has asked him to measure each one and find the circumference and the area of each.

Miguel knows that the circumference is the distance around the edge of the circle. He decides to start with figuring out the circumference of each circle.

He measures the distance across each one.

The first one measures 4 ft. across.

The second measures 5 ft. across.

The third one measures 6 ft across.

Miguel has completed the first part of this assignment. He knows the circumference of each pad. Now he has to figure out the area of each. Miguel isn’t sure how to do this. He can’t remember how to find the area of a circle. Miguel needs some help.

**Now it’s time to help Miguel figure out each area.**

**The first one has a diameter of 4 feet, so it has a radius of 2 ft. Here is the area of the first pad.**

\begin{align*}A&= \pi r^2 \\
A&=3.14(2^2) \\
A&=12.56 \ sq.feet \end{align*}

**The second pad has a diameter of 5 feet, so it has a radius of 2.5 feet.**

\begin{align*}A& = \pi r^2 \\
A& =3.14(2.5^2) \\
A&=19.63 \ sq.feet \end{align*}

**The third pad has a diameter of 6 feet, so it has a radius of 3 feet.**

\begin{align*}A & = \pi r^2 \\
A& = 3.14(3^2) \\
A& = 28.26 \ sq.feet \end{align*}

**Miguel is very pleased with his work. He is sure that his coach will be pleased with his efforts as well!**

### Vocabulary

Here are the vocabulary words in this Concept.

- Circle
- a set of connected points that are equidistant from a center point.

- Diameter
- the distance across the center of a circle.

- Radius
- the distance from the center of the circle to the outer edge.

- Area
- the space inside a two-dimensional figure

### Guided Practice

Here is one for you to try on your own.

Find the area of a circle with a diameter of 10 in.

**Answer**

First, we divide the measurement in half to find the radius.

\begin{align*}10 \div 2 = 5 \ in\end{align*}

Now we use the formula.

\begin{align*}A & = \pi r^2 \\
A& = 3.14(5^2) \\
A&=3.14 (25) \\
A&=78.5 \ square \ inches \end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

- This is a James Sousa video on finding the area of a circle.

### Practice

Directions: Find the area of each circle given the radius or diameter. Round to the nearest hundredth when necessary.

1. \begin{align*}r = 3 \ in\end{align*}

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

3. \begin{align*}r = 4 \ ft\end{align*}

4. \begin{align*}r = 7 \ m\end{align*}

5. \begin{align*}r = 6 \ cm\end{align*}

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

7. \begin{align*}d = 16 \ in\end{align*}

8. \begin{align*}d = 14 \ cm\end{align*}

9. \begin{align*}d = 20 \ in\end{align*}

10. \begin{align*}d = 15 \ m\end{align*}

11. \begin{align*}d = 22 \ cm\end{align*}

12. \begin{align*}d = 24 \ mm\end{align*}

13. \begin{align*}d = 48 \ in\end{align*}

14. \begin{align*}r = 16.5 \ in\end{align*}

15. \begin{align*}r = 25.75 \ in\end{align*}

chord

A line segment whose endpoints are on a circle.circle

The set of all points that are the same distance away from a specific point, called the**.**

*center*circumference

The distance around a circle.diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.pi

(or ) The ratio of the circumference of a circle to its diameter.radius

The distance from the center to the outer rim of a circle.### Image Attributions

Here you'll find the area of a circle given the radius or diameter.

## Concept Nodes:

chord

A line segment whose endpoints are on a circle.circle

The set of all points that are the same distance away from a specific point, called the**.**

*center*circumference

The distance around a circle.diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.pi

(or ) The ratio of the circumference of a circle to its diameter.radius

The distance from the center to the outer rim of a circle.