9.7: Circumference of a Circle
Introduction
Pitchers on Deck
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. Pitchers practice their warm-ups while 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 begins working on his calculations.
While Miguel does this, it is time for you to learn about circumference. By the end of this lesson you will know how to find the circumference of each of the circles just like Miguel does.
What You Will Learn
By the end of this lesson, you will understand how to use the following skills:
- Recognize the formula for the circumference of a circle.
- Find the circumference of a circle given the radius or diameter.
- Find the diameter or radius given the circumference.
- Solve real world problems involving the circumference of circles.
Teaching Time
I. Recognize the Formula for the Circumference of a Circle
In the introduction, the word “circumference” was used. Circumference is a word associated with circles. A circle is a figure whose edge is made of points that are all the same distance from the center. This lesson is all about the circumference of circles. Let’s begin by taking a look at what we mean when we use that word. The circumference of a circle is the distance around the outside edge of a circle.
With other figures, we could find the perimeter of the circle. The perimeter is the distance around a polygon. Circles are not polygons because they are not made up of line segments. When we were finding the perimeter of a polygon, we found the sum of the outer edges.
Circles are quite different. We can’t add up the measurement of the edges, because there aren’t any. To understand circumference, we have to begin by looking at the parts of a circle.
What are the parts of a circle?
You can see here that this is one part of a circle. It is the distance from the center of the circle to the outside edge of the circle. This measurement is called the radius.
You can see that the distance across the center of the circle is called the diameter. The diameter divides the circle into two equal halves. It is twice as long as the radius.
Now we know the basics of circles: the radius, the diameter, and the circumference. Let’s see how we can use these elements in a formula to find the circumference of a circle.
Let’s think about the relationship between the diameter and the circumference.
Think about circles that are drawn on a playground. There are many different sizes and shapes of them. If we were to draw one circle with chalk, that circle would have a diameter and a circumference. If we were to draw a circle around the outside of this other circle, it would have a longer diameter and therefore it would have a larger circumference.
There is a relationship between the size of the diameter and the size of the circumference.
What is this relationship?
It is a proportional relationship that is expressed as a ratio. A ratio simply means that two numbers are related to each other. Circles are special in geometry because this ratio of the circumference and the diameter always stays the same.
Take a look at these two examples.
We can see the ratio when we divide the circumference of a circle by its diameter. No matter how big or small the circle is we will always get the same number. Let’s try it out on the circles above.
\begin{align*}\frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{6.28}{2} \quad = \quad 3.14 && \frac{\text{circumference}}{\text{diameter}} \quad = \quad \frac{12.56}{4} \quad = \quad 3.14\end{align*}
Even though we have two different circles, the result is the same! Therefore the circumference and the diameter always exist in equal proportion, or a ratio, with each other. This relationship is always the same. Whenever we divide the circumference by the diameter, we will always get 3.14. We call this number pi, and we represent it with the symbol \begin{align*}\pi\end{align*}. Pi is actually a decimal that is infinitely long—it has no end. We usually round it to 3.14 to make calculations easier.
Using the equations above, we can write a general formula that shows the relationship between pi, circumference, and diameter.
\begin{align*}\frac{C}{d}= \pi\end{align*}
If we rearrange this formula, we can also use it to find the circumference of a circle when we are given the diameter.
\begin{align*}C = \pi d\end{align*}
We can use this formula to find the circumference of any circle. Remember, the number for \begin{align*}\pi\end{align*} is always the same: 3.14. We simply multiply it by the diameter to get the circumference.
II. Find the Circumference of a Circle Given the Radius or Diameter
Now that you understand the parts of a circle and how the formula for finding the circumference is developed, it is time to put this formula into practice. Let’s look at using it to figure out the circumference of a circle.
Example
Find the circumference of the circle below.
We can see that the diameter of the circle is 8 inches. Let’s put this number into the formula.
\begin{align*}C & = \pi d\\ C & = \pi (8)\\ C & = 25.12 \ in.\end{align*}
The circumference of a circle that has a diameter of 8 inches is 25.12 inches. In other words, if we could unroll the circle into a flat line, it would be 25.12 inches long.
Let’s try another.
Example
What is the area of the circle below?
Again, we know the diameter, so we put it into the formula and solve.
\begin{align*}C & = \pi d\\ C & = \pi (12.7)\\ C & = 39.88 \ m\end{align*}
We can say that a circle with a diameter of 12.7 meters has a circumference of approximately 39.88 meters.
What if we had been given the radius and not the diameter?
Well, the radius is one-half as long as the diameter. So we can multiply the radius by 2 and end up with the same measure as the diameter. Here is how we can alter the formula when given a radius.
\begin{align*}C=2\pi r \end{align*}
Now let’s try it out with an example.
Example
Find the circumference of the following circle.
Now let’s substitute the known information into the formula.
\begin{align*}C&=2 \pi (3) \\ C&=2(3.14)(3) \\ C&=3.14(6) \\ C&=18.84 \ inches \end{align*}
This is our answer.
Now it’s time for you to try a few on your own.
9L. Lesson Exercises
Find the circumference of each circle given the radius or diameter.
- \begin{align*}d = 5 \ in\end{align*}
- \begin{align*}r = 3.5 \ in\end{align*}
- \begin{align*}d = 10 \ ft\end{align*}.
Take a few minutes to check your answers with a neighbor.
Now write down the formulas for finding the circumference of a circle given the radius or the diameter.
III. Find the Diameter or Radius Given the Circumference
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.
Let’s try an example.
Example
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.
Example
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.
9M. Lesson Exercises
Find the missing dimension.
- The circumference is 28.26 inches. What is the diameter?
- The circumference is 21.98 feet. What is the radius?
- The circumference is 34.54 meters. What is the diameter?
Take a few minutes to go over your answers with a peer.
IV. Solve Real World Problems Involving the Circumference of Circles
We have seen that we can apply the formula for finding the circumference of circles to different kinds of situations. Sometimes we need to solve for circumference, but other times we may need to find the diameter or the radius. We can also use this formula when we are given real measurements. Let’s try a few problems involving circles in the real world.
Example
Samuel baked a pie in a 9-inch pie pan. What is the circumference of the pie?
Let’s begin by figuring out what the problem is asking us to find. We need to find the circumference of the pie, so we will use the formula to solve for \begin{align*}C\end{align*}. In order to use the formula, we need to know either the diameter or the radius of the pie. The problem tells us that the diameter of the pie is 9 inches. Let’s put this information into the formula and solve for the circumference.
\begin{align*}C & = \pi d\\ C & = \pi (9)\\ C & = 28.26 \ in.\end{align*}
The circumference of the pie is 28.26 inches.
Good job! Let’s move on.
Example
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?
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.
Now let’s go back to Miguel and the baseball dilemma.
Real–Life Example Completed
Pitchers on Deck
Here is the original problem once again. Reread it and underline any important information.
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. Pitchers practice their warm-ups while 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 begins working on his calculations.
Let’s start with the first on deck pad. It measures 4 feet across. That is our diameter. We multiply that times 3.14 to find the circumference.
\begin{align*}C&=\pi d\\ C&=3.14(4)\\ C&=12.56 \ ft\end{align*}
Next, we work on the on deck pad with a diameter of five feet.
\begin{align*}C&=\pi d\\ C&=3.14(5)\\ C&=15.7 \ ft\end{align*}
Finally we work on the on deck pad with a diameter of 6 feet.
\begin{align*}C&=\pi d\\ C&=3.14(6)\\ C&=18.84 \ ft\end{align*}
Miguel jots down these dimensions. Now he is ready to figure out the area of each on deck pad.
Vocabulary
Here are the vocabulary words found in this lesson.
- 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
Technology Integration
Khan Academy Circumference of a Circle
James Sousa, Determine the Circumference of a Circle
Other Videos:
- http://www.mathplayground.com/mv_circumference.html – This is a Brightstorm video about how to calculate the circumference of a circle.
Time to Practice
Directions: Find the circumference of each circle given the radius or the diameter.
1. radius = 2 in
2. diameter = 4 ft
3. radius = 4.5 in
4. diameter = 8 meters
5. radius = 12 inches
6. diameter = 12 mm
7. radius = 14 mm
8. diameter = 13 feet
9. radius = 10 inches
10. diameter = 7.5 feet
Directions: Given the circumference, find the diameter.
11. Circumference = 15.7 ft.
12. Circumference = 20.41 in
13. Circumference = 21.98 m
14. Circumference = 4.71 cm
15. Circumference = 47.1 ft
Directions: Given the circumference, find the radius.
16. Circumference = 43.96 in
17. Circumference = 15.7 m
18. Circumference = 14.13 in
19. Circumference = 20.41 cm
20. Circumference = 12.56 ft.
Directions: Solve each problem.
21. What is the circumference of a circle whose diameter is 32.5 meters?
22. A circle has a radius of 67 centimeters. What is its circumference?
23. What is the circumference of a circle whose radius is 7.23 feet?
24. What is the diameter of a circle whose circumference is 172.7 inches?
25. A circle has a circumference of 628 centimeters. What is the radius of the circle?
26. The circumference of a circular table is 40.82 feet. What is the radius?
27. 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 five pens?
28. 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?
29. 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?