10.3: Circumference of Circles
Introduction
The Quilting Visitors
Jillian loves having her grandmother at her house for the summer. About halfway through the visit, Jillian’s grandmother receives a phone call from one of her quilting friends. The quilting group that her grandmother is a part of has decided to come for the weekend. Jillian is so excited she can hardly wait.
“My, I hope we have enough room for everyone,” Jillian’s grandmother tells her after getting the phone call.
“How much room does each person need?” Jillian asks, looking at the round table in the dining room.
“Each quilter needs about 2 feet of space to work, and there are six women coming to visit, plus you and me, that makes eight quilters.”
Jillian looks at the table in the dining room again. It is a circle with a diameter of 6 feet across.
How can Jillian figure out if everyone will fit at the table?
Jillian will need to figure out the distance around the table or the circumference of the circle of the table. Then she can figure out if all 8 people will have 2 feet to work.
You will have to figure this out too. In this lesson you will learn all about the circumference of a circle. Pay close attention and then you’ll be able to help Jillian with her problem.
What You Will Learn
By the end of this lesson you will learn the following skills:
- Identify ratio of circumference to diameter as pi.
- Find the circumference of circles given diameter or radius.
- Find diameter or radius of circles given circumference.
- Solve real world problems involving circumference of circles.
Teaching Time
I. Identify Ratio of Circumference to Diameter as Pi
To work with circles, we first need to review the parts of a circle. Let’s begin there.
We can measure several key parts of a circle. We can measure the distance across the center of the circle. This distance is called the diameter of the circle. Here is a picture of the diameter.
We can measure the distance from the center of the circle to the outer edge. This distance is called the radius. Notice that the radius is one-half of the measure of the diameter. Here is a picture of the radius.
We can measure the perimeter of the circle too. This distance is called the circumference of the circle.
To understand things about circles, let’s travel back in time. Here is our time machine.
We have traveled all the way back to a time when the Greeks were discovering all sorts of things about mathematics. They were puzzled by mathematics and by the relationships between different measurements and geometry. The Greeks were famous for investigating ratios and proportions. When they studied different things, they knew that there was a connection between shapes and their measurements. Some of the Greeks thought a lot about circles.
Although the Babylonians had been investigating circles too, it was a Greek man named Archimedes who is credited with figuring out that there is a relationship between the diameter and circumference of a circle.
Archimedes discovered that if you take the distance across the circle and stretch it around the circumference, that the length of the diameter will go around the circle 3 and a bit more times.
Let’s say that the diameter of this circle is 5 cm, in that case the circumference of the circle is three and a little more times the 5 cm, or a little less than 16 cm.
We say that the ratio of the diameter to the circumference is pi. We use the number 3.14 for pi because the actual ratio is a non-terminating decimal, which means it cannot be written precisely as a numeral since the decimal places never end or form a pattern. However, using two decimal places for pi works for estimating the circumference of the circle.
Here is the symbol for pi. When you see this symbol, you can use 3.14 in your arithmetic.
Think about what you have learned and answer these questions.
- Who was the first person to figure out the relationship between the diameter and the circumference?
- What is the distance across the circle called?
- What is the distance around the circle called?
Take a few minutes to check your answers with a friend.
II. Find the Circumference of a Circle Given the Diameter or Radius
Now that you know about the relationship between the diameter and circumference of a circle, we can work on figuring out the circumference using a formula and pi.
To figure out the circumference of the circle, we multiply the diameter of the circle times pi or 3.14.
\begin{align*}C = d\pi\end{align*}
Remember, whenever you see the symbol for pi, you substitute 3.14 in when multiplying.
Let’s look at an example.
Example
Find the circumference.
The diameter of the circle is 6 inches. We can substitute this given information into our formula and solve for the circumference of the circle.
\begin{align*}C & = d\pi\\ C & = 6(3.14)\\ C & = 18.84 \ inches\end{align*}
Our answer is 18.84 inches.
What if we have been given the radius and not the diameter?
Let’s look at an example.
Example
Find the circumference.
Remember that the radius is one-half of the diameter. You can solve this problem in two ways.
- Double the radius right away and then use the formula for diameter to find the circumference. OR
- Use this formula:
\begin{align*}C = 2\pi r\end{align*}
Let’s use the formula to find the circumference of the circle.
\begin{align*}C & = 2(3.14)(4)\\ C & = 3.14(8)\\ C & = 25.12 \ cm\end{align*}
Our answer is 25.12 cm.
Practice a few of these on your own. Find the circumference given the radius or diameter.
- \begin{align*}d = 5 \ in\end{align*}
- \begin{align*}r = 3 \ in\end{align*}
- \begin{align*}d = 2.5 \ cm\end{align*}
Take a few minutes to check your work with a peer. Did you get stuck when the diameter was a decimal too? Remember the rules for multiplying decimals and you will be all set!
III. Find the Diameter or Radius of a Circle Given Circumference
What happens if you are given the circumference but not the radius or the diameter? Can you still solve for one or the other?
Working in this way is a bit tricky and will require us to play detectives once again. You will have to work backwards to figure out the radius and/or the diameter when given only the circumference.
Let’s look at an example and see how this works.
Example
Find the diameter of a circle with a circumference of 21.98 feet.
To work on this problem, we will need our formula for finding the circumference of a circle.
\begin{align*}C = \pi d\end{align*}
Next, we fill in the given information.
\begin{align*}21.98 = (3.14)d\end{align*}
To solve this problem we need to figure out what times 3.14 will give us 21.98. To do this, we divide 21.98 by 3.14.
\begin{align*}{3.14 \overline{ ) {21.98 \;}}}\end{align*}
Remember dividing decimals? First, we move the decimal point two places to make our divisor a whole number. Then we can divide as usual.
\begin{align*}& \overset{ \qquad \ \quad 7}{314 \overline{ ) {2198}}}\end{align*}
The diameter of this circle is 7 feet.
How could we figure out the radius once we know the diameter?
We can figure out the radius by dividing the diameter in half. The radius is one-half the measure of the diameter.
7 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} 3.5
The radius of the circle is 3.5 feet.
Try a few of these on your own. Figure out the diameter and the radius given the circumference of the circle.
- \begin{align*}C = 31.4 \ m\end{align*}
- \begin{align*}C = 28.26 \ in\end{align*}
Check your answers with a partner. Is your work accurate? Did you find the measure of both the radius and the diameter?
Real Life Example Completed
The Quilting Visitors
Here is the original problem once again. Reread it and underline any important information.
Jillian loves having her grandmother at her house for the summer. About halfway through the visit, Jillian’s grandmother receives a phone call from one of her quilting friends. The quilting group that her grandmother is a part of has decided to come for the weekend. Jillian is so excited she can hardly wait.
“My, I hope we have enough room for everyone,” Jillian’s grandmother tells her after getting the phone call.
“How much room does each person need?” Jillian asks, looking at the round table in the dining room.
“Each quilter needs about 2 feet of space to work, and there are six women coming to visit, plus you and me, that makes eight quilters.”
Jillian looks at the table in the dining room again. It is a circle with a diameter of 6 feet across.
How can Jillian figure out if everyone will fit at the table?
Jillian will need to figure out the distance around the table or the circumference of the circle of the table. Then she can figure out if all 8 people will have 2 feet to work.
First, let’s look at a picture of the table in Jillian’s house.
Diameter = 6 feet
Next, Jillian needs to find the circumference. Here is the formula we can use to help her out.
\begin{align*}C & = \pi d\\ C & = (3.14)(6)\\ C & = 18.84 \ feet\end{align*}
Next, Jillian needs to figure out if all eight people will fit given the circumference and the fact that each person needs two feet of space.
If we divide the circumference by the two feet of space, we will know if 8 people can fit around the table.
\begin{align*}& \overset{\quad \ 9.42}{2 \overline{ ) {18.84}}}\end{align*}
Given this work, 9 and almost one-half people can fit at the table.
Jillian shows this work to her grandmother, who is very pleased. The entire group can work together and have a little extra space left over.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Circumference
- the measure of the distance around the outside edge of a circle.
- Diameter
- the measure of the distance across the center of a circle.
- Radius
- the measure of the distance half-way across the circle. It is the measure from the center to the outer edge. The radius is also half the length of the diameter.
- Pi
- the ratio of the diameter to the circumference, 3.14
- Archimedes
- a Greek mathematician and philosopher who identified 3.14 as pi.
Technology Integration
Khan Academy, Circumference of a Circle
James Sousa, Example of Determining the Circumference of a Circle
Other Videos:
- http://www.mathplayground.com/mv_circumference.html – This is a Brightstorm video that is a basic blackboard video about finding the circumference of a circle.
Time to Practice
Directions: Find the circumference of each circle given the diameter.
1. \begin{align*}d = 5 \ in\end{align*}
2. \begin{align*}d = 8 \ in\end{align*}
3. \begin{align*}d = 9 \ cm\end{align*}
4. \begin{align*}d = 3 \ cm\end{align*}
5. \begin{align*}d = 10 \ ft\end{align*}
6. \begin{align*}d = 15 \ ft\end{align*}
7. \begin{align*}d = 11 \ m\end{align*}
8. \begin{align*}d = 13 \ ft\end{align*}
9. \begin{align*}d = 17 \ ft\end{align*}
10. \begin{align*}d = 20 \ in\end{align*}
Directions: Find the circumference of each circle given the radius.
11. \begin{align*}r = 2.5 \ in\end{align*}
12. \begin{align*}r = 4 \ in\end{align*}
13. \begin{align*}r = 4.5 \ cm\end{align*}
14. \begin{align*}r = 1.5 \ cm\end{align*}
15. \begin{align*}r = 5 \ ft\end{align*}
16. \begin{align*}r = 7.5 \ ft\end{align*}
17. \begin{align*}r = 5.5 \ m\end{align*}
18. \begin{align*}r = 6.5 \ ft\end{align*}
19. \begin{align*}r = 8.5 \ ft\end{align*}
20. \begin{align*}r = 10 \ in\end{align*}
Directions: Answer the following questions using what you have learned.
21. What comparison can you make between the answers in 1 – 10 and the answers in 10 – 20?
22. Why do you think this is?
23. What is the distance half way across a circle called?
24. What is the symbol for pi?
25. What is the mathematical number used to represent pi?
26. What is the distance across the center of a circle called?
27. What is the distance around a circle called?
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