10.4: Area of Circles
Introduction
The Quilting Survey
Jillian loves quilting. At first, she thought that she would love it because it is something that she could do with her grandmother, but now she is sure that she actually really loves the quilting itself. Jillian loves creating something with her hands and seeing the finished project.
Jillian asked her grandmother how long she had been quilting and her grandmother told her that she had been quilting for a long time, long before Jillian was even born. Jillian began to wonder how many other people in the world quilt.
During a trip to the library, Jillian used the computer to do some research. She found that the number of people quilting in the United States has increased significantly from 2006 to 2009. Quilters.com completed a survey and here are their results.
In 2006, 21.3 million people were quilting. That means that about 13% of all Americans were making quilts.
In 2009, 27 million people were quilting. That means that about 17% of all Americans were quilting.
That is an increase of 4% in just three years! It might not seem like much, but it is a significant increase!
Jillian wants to show her grandmother the results of the survey. She has decided to create a picture of the data to show how the information has changed. To do this, she is going to create a circle graph.
She will create two circle graphs, one for 2006 and one for 2009.
Do you know how to do this? During this lesson you will continue learning about circles. At the end of the lesson, you will see how Jillian used a circle to display the quilting data.
What You Will Learn
- Find areas of circles given radius or diameter.
- Find radius or diameter of circles given area.
- Find areas of combined figures involving parts of circles.
- Display real-world data using circle graphs.
Teaching Time
I. Find Areas of Circles Given Radius or Diameter
In the last Concept you learned about the parts of a circle and about finding the circumference of a circle. This Concept is going to focus on the inside of the circle. The inside of a circle is called the area of the circle.
What is the area of a circle?
Remember back to working with quadrilaterals? The area of the quadrilateral is the surface or space inside the quadrilateral. Well, the area of a circle is the same thing. It is the area inside the circle that we are measuring.
In this picture, the area of the circle is yellow.
How do you measure the area of a circle?
To figure out the area of a circle, we are going to need a couple of different measurements. The first one is pi. We will need to use the numerical value for pi, or 3.14, to represent the ratio between the diameter and the circumference.
The next measurement we need to use is the radius. Remember that the radius is the distance from the center of a circle to the edge, or is 1/2 of the circle's diameter.
We calculate the area of a circle by multiplying the radius squared (multiplied by itself) by pi (3.14).
Here is the formula.
\begin{align*}A = \pi r^2\end{align*}
Here is an example.
Example
Find the area of the circle.
Next, we use our formula and the given information.
\begin{align*}A & = \pi r^2\\ A & = (3.14)(3^2)\\ A & = (3.14)(9)\\ A & = 28.26 \ mm^2\end{align*}
What about if you have been given the diameter and not the radius?
If you have been given the diameter and not the radius, you can still figure out the area of the circle. You start by dividing the measure of the diameter in half since the radius is one-half the measure of the diameter.
Then you use the formula and solve for area.
Let’s look at an example.
Example
Notice that the diameter is 6 inches. We can divide this in half to find the radius.
6 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} 3 inches \begin{align*}=\end{align*} radius
Next, we substitute the given values into the formula and solve for the area of the circle.
\begin{align*}A & = \pi r^2\\ A & = (3.14)(3^2)\\ A & = (3.14)(9)\\ A & = 28.26 \ in^2\end{align*}
Now it’s time for you to try a few on your own. Find the area of the circle using the given radius or diameter.
- \begin{align*}r = 4 \ cm\end{align*}
- \begin{align*}r = 8 \ cm\end{align*}
- \begin{align*}d = 4 \ in\end{align*}
Take a few minutes to check your work. Did you remember to label your answer using square inches or centimeters?
II. Find the Radius or Diameter of Circles Given Area
Now that you know how to find the area of a circle given a radius or diameter, we can work backwards and use the area to find the radius or the diameter.
It is time to use your detective skills again!!
Now, let’s look at an example.
Example
The area of the circle is \begin{align*}153.86 \ in^2\end{align*}, what is the radius? What is the diameter?
This problem asks for you to figure out two different things. First, let’s find the radius and then we can use that measure to figure out the diameter.
Let’s begin by using the formula for finding the area of a circle.
\begin{align*}A & = \pi r^ 2\\ 153.86 & = (3.14)r^2\end{align*}
We substituted in the given information. We know the area, and we know that the measure for pi is 3.14. Next, we can divide the area by pi. This will help to get us one step closer to figuring out the radius.
\begin{align*}{3.14 \overline{ ) {153.86}}}\end{align*}
Remember, when you divide decimals, we move the decimal two places in the divisor and the dividend. Here is our new division problem.
\begin{align*}{314 \overline{ ) {15386}}}\end{align*}
Yes. It is a large number to divide, but don’t let that stop you. Just work through it step by step and you will be able to find the correct answer!
\begin{align*}& \overset{\qquad \ \quad 49}{314 \overline{ ) {15386}}}\\ & \quad \underline{-1256 \ \ }\\ & \qquad \ 2826\\ & \quad \ \ \underline{-2826}\\ & \qquad \qquad 0\end{align*}
So far, our answer is 49, but that is not the radius.
\begin{align*}49 = r^2\end{align*}
We need to figure out which number times itself is equal to 49.
Now we know that the radius is 7 because 7 \begin{align*}\times\end{align*} 7 \begin{align*}=\end{align*} 49.
The radius is 7 inches.
What is the diameter?
The measure of the radius is one-half the measure of the diameter. If the radius is 7, then the diameter is double that.
7 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 14
The diameter is 14 inches.
Warning! Working backwards is tricky! Be sure that you take your time when working through problems!
Try a few of these on your own.
- If the area of a circle is \begin{align*}12.56 \ cm^2\end{align*}. What is the radius? What is the diameter?
- If the area of a circle is \begin{align*}113.04 \ m^2\end{align*}. What is the radius? What is the diameter?
Check your answers with a partner. Are your answers accurate?
III. Find Areas of Combined Figures Involving Parts of Circles
Sometimes, there will be figures that aren’t quadrilaterals and they aren’t circles either, they are combined figures. A combined figure is a figure that is made up of more than one type of polygon. You can still figure out the area of combined figures, but you will have to think about how to do it!!
Let’s look at an example.
Example
What is the area of the figure?
To solve this problem, you first have to look at which figures have been combined. Here you have one-half of a circle and a rectangle.
We will need to figure out the area of the rectangle, the area of half of the circle and then add the two areas.
This will give us the area of the combined figure.
Let’s start with the rectangle.
\begin{align*}A = lw\end{align*}
The length of the rectangle is 6 inches. The width of the rectangle is 3 inches.
\begin{align*}A & = (6)(3)\\ A & = 18 \ in^2\end{align*}
Next, we find the area of the circle. We can start by noticing that the length of the rectangle is also the diameter of the circle. The diameter of the circle is 6 inches. We can start by figuring out the area of one whole circle and then divide that in half for the area of half of the circle.
If the diameter of the circle is 6 inches, then the radius is 3 inches. Remember that the radius is one-half of the diameter.
\begin{align*}A & = \pi r^2\\ A & = (3.14)(3^2)\\ A & = 28.26 \ in^2\end{align*}
This is the area of the whole circle. Our figure only has half of a circle, so we divide this in half.
28.26 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} 14.13 in
Now we combine the area of the rectangle with the area of the half circle. This will equal the area of the entire figure.
18 + 14.13 = 32.13
The area of the figure is \begin{align*}32.13 \ in^2\end{align*}.
Try one of these on your own. Remember, separate the figure and find the area of the parts, then combine the areas.
Take a few minutes to check your work with a friend, then move on to the next section.
IV. Display Real-World Data Using Circle Graphs
In our introduction problem, Jillian wants to display her quilting data in a circle graph. We can use circle graphs to display real-world data. In fact, we do it all the time.
What is a circle graph?
A circle graph is a visual way to display data using circles and parts of a circle.
A circle graph uses a circle to indicate 100%. The entire circle represents 100% and each section of a circle represents some part out of 100.
You can see here that this circle graph is divided into five sections. Each section represents a part of a whole.
Back in Chapter Two, we looked at the spending habits of a teenager. Here you can see that 50% of his money went into savings. 40% of his money was spent on food and that 10% of his money went to baseball cards.
When Jillian creates her circle graphs, she will be able to create ones that show how quilting has grown from 2006 to 2009. Let’s go and revisit that introductory problem now.
Real Life Example Completed
The Quilting Survey
Here is the original problem once again. Reread it and underline any important information.
Jillian loves quilting. At first, she thought that she would love it because it is something that she could do with her grandmother, but now she is sure that she actually really loves the quilting itself. Jillian loves creating something with her hands and seeing the finished project.
Jillian asked her grandmother how long she had been quilting and her grandmother told her that she had been quilting for a long time, long before Jillian was even born. Jillian began to wonder how many other people in the world quilt.
During a trip to the library, Jillian used the computer to do some research. She found that the number of people quilting in the United States has increased significantly from 2006 to 2009. Quilters.com completed a survey and here are their results.
In 2006, 21.3 million people were quilting. That means that about 13% of all Americans were making quilts.
In 2009, 27 million people were quilting. That means that about 17% of all Americans were quilting.
That is an increase of 4% in just three years! It might not seem like much, but it is a significant increase!
Jillian wants to show her grandmother the results of the survey. She has decided to create a picture of the data to show how the information has changed. To do this, she is going to create a circle graph.
She will create two circle graphs, one for 2006 and one for 2009.
Let’s look at Jillian’s data.
The first circle graph will show that 13% out of 100% of people were quilting in 2006. Here is the circle graph.
The second circle graph shows that in 2009, the number of people quilting increased to 17% out of 100%.
Now Jillian has two circle graphs that she can share with her grandmother.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Area
- the surface or space of the figure inside the perimeter.
- Radius
- the measure of the distance halfway across a circle.
- Diameter
- the measure of the distance across a circle
- Squaring
- uses the exponent 2 to show that a number is being multiplied by itself. \begin{align*}3^2 = 3 \times 3\end{align*}
- Pi
- the ratio of the diameter to the circumference. The numerical value of pi is 3.14.
Technology Integration
Khan Academy, Area of a Circle
James Sousa, Example of Determining the Area of a Circle
Other Videos:
- http://www.mathplayground.com/mv_area_circles.html – This is a basic blackboard video on finding the area of a circle by Brightstorm.
Time to Practice
Directions: Find the area of the following circles given the radius.
1. \begin{align*}r = 4 \ in\end{align*}
2. \begin{align*}r = 5 \ cm\end{align*}
3. \begin{align*}r = 8 \ in\end{align*}
4. \begin{align*}r = 2 \ cm\end{align*}
5. \begin{align*}r = 7 \ m\end{align*}
6. \begin{align*}r = 9 \ in\end{align*}
7. \begin{align*}r = 10 \ ft\end{align*}
8. \begin{align*}r = 11 \ cm\end{align*}
9. \begin{align*}r = 20 \ ft\end{align*}
10. \begin{align*}r = 30 \ miles\end{align*}
Directions: Find the area of the following circles given the diameter.
11. \begin{align*}d = 10 \ in\end{align*}
12. \begin{align*}d = 12 \ m\end{align*}
13. \begin{align*}d = 14 \ cm\end{align*}
14. \begin{align*}d = 16 \ ft\end{align*}
15. \begin{align*}d = 18 \ in\end{align*}
16. \begin{align*}d = 22 \ ft\end{align*}
17. \begin{align*}d = 24 \ cm\end{align*}
18. \begin{align*}d = 28 \ m\end{align*}
19. \begin{align*}d = 30 \ m\end{align*}
20. \begin{align*}d = 36 \ ft\end{align*}
Directions: Find the area of each of the following.
21.
22.
23.
24.