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Diameter or Radius of a Circle Given Circumference

C = πd; C = 2πr

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Diameter or Radius of a Circle Given Circumference

Let’s Think About It

Credit: Jason Eppink
Source: https://www.flickr.com/photos/jasoneppink/5440728084
License: CC BY-NC 3.0

Rebekah’s family went to a circular corn maze and spent a couple hours trying to find their way out. The distance around the outside of the maze was posted as 1 mile, or 5,280 feet. Rebekah wondered how far she would walk if she were able to cut right through the middle.

In this concept, you will learn to find the diameter and radius of a circle if you are given the circumference.

Guidance

The formula for the circumference of a circle, \begin{align*}c= \pi d\end{align*} or \begin{align*}c=2 \pi r\end{align*}, can also be used to find the diameter or radius of a circle.

Let's look at an example.

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

First, write down the formula.

\begin{align*}c=\pi d\end{align*}

Next, fill in the values that you know.

\begin{align*}20.72 \ m=3.14 \ d\end{align*}

Then, divide both sides of the equation by 3.14

\begin{align*}6.6 = d\end{align*}

The answer is the diameter is 6.6 meters.

You can check your answer by substituting your answer in the original equation.

\begin{align*}\begin{array}{rcl} c &=& \pi d \\ 20.72 &=& (3.14)(6) \\ 20.72 &=& 20.72 \end{array}\end{align*}The answer checks.

Guided Practice

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

First, write down the formula.

\begin{align*}c=2 \pi r\end{align*}

Next, fill in the values that you know.

\begin{align*}147.58=2(3.14)r\end{align*}

Then, perform the calculations necessary to isolate \begin{align*}r\end{align*}.

\begin{align*}\begin{array}{rcl} 147.58 &=& 6.28 \ r \\ 147.58 &=& 6.28 \ r \\ 23.5 &=& r \end{array}\end{align*}The answer is \begin{align*}r = 23.5 \end{align*} yards.

Examples

Example 1

The circumference is 28.26 inches. What is the diameter?

First, write down the formula.

\begin{align*}c= \pi d\end{align*}

Next, fill in the values that you know.

\begin{align*}28.26=3.14d\end{align*}

Then, divide both sides of the equation by 3.14

\begin{align*}9 = d\end{align*}

The answer is the diameter is 9 inches.

Example 2

The circumference is 21.98 feet. What is the radius?

First, write down the formula.

\begin{align*}c=2 \pi r\end{align*}

Next, fill in the values that you know.

\begin{align*}21.98=2(3.14)r\end{align*}

Then, perform the calculations necessary to isolate \begin{align*}r\end{align*}.

\begin{align*}\begin{array}{rcl} 21.98 &=& 6.28 \ r \\ 3.5 &=& r \end{array}\end{align*}

The answer is \begin{align*}r = 3.5\end{align*} feet.

Follow Up

Credit: Angela Severn
Source: https://www.flickr.com/photos/43072779@N06/4026852130
License: CC BY-NC 3.0

Remember Rebekah at the corn maze with a 5,280 ft circumference?

She wondered how far she would have to walk if she were able to cut right across.

First, recognize that you are solving for the diameter, the straight-line distance through the middle of the circle from one point on the circumference to another.

Then, write the formula.

\begin{align*}c= \pi d\end{align*}

Next, fill in the values that you know.

\begin{align*}5,280=(3.14)d\end{align*}

Then, perform the calculations necessary to isolate \begin{align*}d\end{align*}.

\begin{align*}\begin{array}{rcl} 1,681.528662 &=& d \\ 1,681.5 \ \text{feet} &=& d. \end{array}\end{align*}

The answer is the \begin{align*}\text{diameter} = 1,681.5 \end{align*} feet. Rebekah would have to walk 1,681.5 feet if she were able to walk the diameter of the corn maze.

Video Review

https://www.youtube.com/watch?v=sHtsnC2Mgnk

Explore More

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

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.

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?

Vocabulary

\pi

\pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.
Circle

Circle

A circle is the set of all points at a specific distance from a given point in two dimensions.
Circumference

Circumference

The circumference of a circle is the measure of the distance around the outside edge of a circle.
Perimeter

Perimeter

Perimeter is the distance around a two-dimensional figure.
Pi

Pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Image Attributions

  1. [1]^ Credit: Jason Eppink; Source: https://www.flickr.com/photos/jasoneppink/5440728084; License: CC BY-NC 3.0
  2. [2]^ Credit: Angela Severn; Source: https://www.flickr.com/photos/43072779@N06/4026852130; License: CC BY-NC 3.0

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