# 9.16: Diameter or Radius of a Circle Given Circumference

**At Grade**Created by: CK-12

Have you ever been swimming? Do you know anyone whose favorite sport is swimming?

Miguel loves baseball, but Terrence loves swimming. Terrence swam around the edge of a circular pool and found that it took him 176 strokes to swim one complete time around the pool.

About how many strokes will it take him to swim across the pool?

You can use what you have already learned about circumference to figure this out, but you will need to think of it in a different way.

Do you know how to do this?

**This Concept will teach you exactly how to figure out this problem.**

### Guidance

**Sometimes a problem will give us the circumference of a circle and ask us to find either its diameter or its radius. We can still use the formula for circumference. All we have to do is put the information we have into the appropriate place in the formula and solve for the unknown quantity.**

A circle has a circumference of 20.72 m. What is its diameter?

**In this problem, we are given the circumference and we need to find the diameter. We put these numbers into the formula and solve for \begin{align*}d\end{align*}.**

\begin{align*}C & = \pi d\\ 20.72 & = \pi d\\ 20.72 \div \pi &= d\\ 6.6 & = d\end{align*}

**By solving for \begin{align*}d\end{align*}, we have found that the diameter of the circle is 6.6 meters.**

Let’s check our calculation to be sure. We can check by putting the diameter into the formula and solving for the circumference:

\begin{align*}C & = \pi d\\ C & = \pi (6.6)\\ C & = 20.72 \ m \end{align*}

**We know the circumference is 20.72 meters, so our calculation is correct.**

The circumference of a circle is 147.58 yards. Find its radius.

**Again, we have been given the circumference. Read carefully! This time we need to find the radius, not the diameter. We can use the formula for radii and solve for \begin{align*}r\end{align*}.**

\begin{align*}C & = 2 \pi r\\ 147.58 & = 2 \pi r\\ 147.58 & = 6.28r\\ 147.58 \div 6.28 & = r\\ 23.5 \ yd & = r\end{align*}

**We have found that the circle has a radius of 23.5 yards.**

This time let’s try checking our work by using the other formula to find the diameter. Remember the diameter is twice the length of the radius.

\begin{align*}C & = \pi d\\ 147.58 & = \pi d\\ 147.58 \div \pi & = d\\ 47 \ yd & = d\end{align*}

**We have found that the diameter of the circle is 47 yards. The radius must be half this length, or \begin{align*}47 \div 2 = 23.5\end{align*} yards.**

**Our calculation is correct!**

Whenever we are given the circumference, we can use the formula to solve for the diameter or the radius. The number for *pi* always stays the same, so we only need one piece of information about a circle to find the other measurement.

Find the missing dimension.

#### Example A

**The circumference is 28.26 inches. What is the diameter?**

**Solution: \begin{align*}9\end{align*} inches**

#### Example B

**The circumference is 21.98 feet. What is the radius?**

**Solution: \begin{align*}3.5\end{align*} feet**

#### Example C

**The circumference is 34.54 meters. What is the diameter?**

**Solution: \begin{align*}11\end{align*} meters**

Here is the original problem once again.

Miguel loves baseball, but Terrence loves swimming. Terrence swam around the edge of a circular pool and found that it took him 176 strokes to swim one complete time around the pool.

About how many strokes will it take him to swim across the pool?

You can use what you have already learned about circumference to figure this out, but you will need to think of it in a different way.

Do you know how to do this?

To figure this out, first let's write down the formula for finding the circumference of a circle.

\begin{align*}C = (3.14)d\end{align*}

We have been given the number of strokes that it took Terrence to get around the pool. This is the circumference.

\begin{align*}176 = (3.14)d\end{align*}

Now we want to figure out the number of strokes it will take him to get across the pool. This is the diameter.

Let's solve the problem by dividing.

\begin{align*}176 \div 3.14 = d\end{align*}

\begin{align*}d = 56\end{align*}

**It will take Terrence 56 strokes to cross the pool.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Circumference
- the distance around the outside edge of a circle.

- Perimeter
- the distance around the edge of a polygon.

- Circle
- a series of connected points equidistant from a center point.

- Radius
- the distance halfway across a circle.

- Diameter
- the distance across the center of circle.

- Pi
- the ratio of circumference to diameter, 3.14

- Ratio
- a comparison between two quantities

### Guided Practice

Here is one for you to try on your own.

Maria wants to paste some ribbon around a circular mirror to make a border. The mirror is 40 inches across. If the ribbon is sold by the inch and costs $0.15 per inch, how much will Maria need to spend to buy enough ribbon?

**Answer**

What is this problem asking us to find? We need to find how much money Maria will spend on the ribbon.

In order to determine this, we first need to know how much ribbon is necessary to go around the mirror. Therefore we need to find the circumference of the mirror. We know that it is 40 inches across. Is this the radius or the diameter? It is the diameter, so we can put this information into the formula and solve for the circumference.

\begin{align*}C & = \pi d\\ C & = \pi (40)\\ C & = 125.6 \ in.\end{align*}

The circumference of the mirror is 125.6 inches, so Maria will need 125.6 inches of ribbon to put around it.

**We’re not done yet, however. Remember, we need to find how much money she will spend to buy the ribbon.**

Because the ribbon is sold by the inch, Maria will need to buy 126 inches. We know that it is sold at $0.15 an inch, so we simply multiply the number of inches by the cost per inch.

\begin{align*}126 \ inches \times \$ 0.15 \ per \ inch = \$ 18.90\end{align*}

**Maria will need to spend $18.90 in order to buy enough ribbon to go around her mirror.**

### Video Review

Here is a video for review.

- This is a James Sousa video on determining the circumference of a circle.

### Practice

Directions: Given the circumference, find the diameter.

1. Circumference = 15.7 ft.

2. Circumference = 20.41 in

3. Circumference = 21.98 m

4. Circumference = 4.71 cm

5. Circumference = 47.1 ft

Directions: Given the circumference, find the radius.

6. Circumference = 43.96 in

7. Circumference = 15.7 m

8. Circumference = 14.13 in

9. Circumference = 20.41 cm

10. Circumference = 12.56 ft.

Directions: Solve each problem.

11. What is the circumference of a circle whose diameter is 32.5 meters?

12. A circle has a radius of 67 centimeters. What is its circumference?

13. What is the circumference of a circle whose radius is 7.23 feet?

14. What is the diameter of a circle whose circumference is 172.7 inches?

15. A circle has a circumference of 628 centimeters. What is the radius of the circle?

16. The circumference of a circular table is 40.82 feet. What is the radius of the rug?

17. Workers at the zoo are building five circular pens for the elephants. Each pen has a diameter of 226 meters. How much fence will the workers need in order to surround all three pens?

18. Mrs. Golding has a circular mirror with a frame around it. The frame is 4 inches wide. If the diameter of the mirror and the frame together is 48 inches, what is the circumference of just the mirror?

### Image Attributions

Here you'll learn to find the diameter or radius given the circumference.