<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

10.10: Area of a Circle

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated15 minsto complete
%
Progress
Practice Area of a Circle
Practice
Progress
Estimated15 minsto complete
%
Practice Now
Turn In

What if you wanted to figure out the area of a circle with a radius of 5 inches? 

Area of a Circle 

Recall that π is the ratio between the circumference of a circle and its diameter. We are going to use the formula for circumference to derive the formula for area.

First, take a circle and divide it up into several wedges, or sectors. Then, unfold the wedges so they are all on one line, with the points at the top.

Notice that the height of the wedges is r, the radius, and the length is the circumference of the circle. Now, we need to take half of these wedges and flip them upside-down and place them in the other half so they all fit together.

Now our circle looks like a parallelogram. The area of this parallelogram is A=bh=πrr=πr2.

The formula for the area of a circle is A=πr2 where r is the radius of the circle.

 

 

 

Finding the Area 

Find the area of a circle with a diameter of 12 cm.

If the diameter is 12 cm, then the radius is 6 cm. The area is A=π(62)=36π cm2.

Finding the Radius 

If the area of a circle is 20π, what is the radius?

Work backwards on this problem. Plug in the area and solve for the radius.

20π20r=πr2=r2=20=25

Just like the circumference, we will leave our answers in terms of π, unless otherwise specified. In Example 2, the radius could be ±25, however the radius is always positive, so we do not need the negative answer.

Calculating the Area of a Circle Inscribed in a Square 

A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle?

The diameter of the circle is the same as the length of a side of the square. Therefore, the radius is half the length of the side, or 5 cm.

A=π52=25π cm

 

 

 

 

Earlier Problem Revisited

A circle with a radius of 5 inches has area π52=25π in2.

Examples

Example 1

Find the area of the shaded region from Example C.

The area of the shaded region would be the area of the square minus the area of the circle.

A=10225π=10025π21.46 cm2

Example 2

Find the diameter of a circle with area 36π.

First, use the formula for the area of a circle to solve for the radius of the circle.

A36π36r=πr2=πr2=r2=6

If the radius is 6 units, then the diameter is 12 units.

Example 3

Find the area of a circle with diameter 20 inches.

If the diameter is 20 inches that means that the radius is 10 inches. Now we can use the formula for the area of a circle. A=π(10)2=100π in2 .

Interactive Practice

 

Review 

Fill in the following table. Leave all answers in terms of π.

radius area diameter
1. 2
2. 16π
3. 10
4. 24
5. 9
6. 90π
7. 35
8. 7π
9. 60
10. 36π

Find the area of the shaded region. Round your answer to the nearest hundredth.

  1. Carlos has 400 ft of fencing to completely enclose an area on his farm for an animal pen. He could make the area a square or a circle. If he uses the entire 400 ft of fencing, how much area is contained in the square and the circle? Which shape will yield the greatest area?

Review (Answers)

To view the Review answers, open this PDF file and look for section 10.10. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Vocabulary

chord

A line segment whose endpoints are on a circle.

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

pi

(or \pi) The ratio of the circumference of a circle to its diameter.

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Grades:
Date Created:
Jul 17, 2012
Last Modified:
Aug 02, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.GEO.840.L.2
Here