10.10: Area of a Circle
What if you wanted to figure out the area of a circle with a radius of 5 inches? After completing this Concept, you'll be able to answer questions like this.
Watch This
CK12 Foundation: Chapter10AreaofaCircleA
Guidance
Recall that
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
Now our circle looks like a parallelogram. The area of this parallelogram is
To see an animation of this derivation, see http://www.rkm.com.au/ANIMATIONS/animationCircleAreaDerivation.html, by Russell Knightley.
The formula for the area of a circle is
Example A
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
Example B
If the area of a circle is
Work backwards on this problem. Plug in the area and solve for the radius.
Just like the circumference, we will leave our answers in terms of
Example C
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.
Watch this video for help with the Examples above.
CK12 Foundation: Chapter10AreaofaCircleB
Concept Problem Revisited
A circle with a radius of 5 inches has area
Vocabulary
A circle is the set of all points that are the same distance away from a specific point, called the center. A radius is the distance from the center to the outer rim of the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. Area is the amount of space inside a figure and is measured in square units.
Guided Practice
1. Find the area of the shaded region from Example C.
2. Find the diameter of a circle with area
3. Find the area of a circle with diameter 20 inches.
Answers:
1. The area of the shaded region would be the area of the square minus the area of the circle.
2. First, use the formula for the area of a circle to solve for the radius of the circle.
If the radius is 6 units, then the diameter is 12 units.
3. 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.
Practice
Fill in the following table. Leave all answers in terms of
radius  area  diameter  

1.  2  
2. 


3. 


4. 


5.  9  
6. 


7. 


8. 


9.  60  
10. 

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

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 ) The ratio of the circumference of a circle to its diameter. 
Image Attributions
Here you'll learn how to calculate the area of a circle.