1. 11.2 Area of Circles and Sectors
Geometry

2. Geometry 11.2 Areas of Circles and Sectors
Goals
Find the area of a circle.
Find the area of a circle sector.
Solve problems using areas.
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

3. Geometry 11.2 Areas of Circles and Sectors
The Area of a Circle
Finding the area of a circle is harder than you might think.
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Geometry 11.2 Areas of Circles and Sectors

4. Geometry 11.2 Areas of Circles and Sectors
Method 1: Archimedes
Start with a circle.
Inscribe a hexagon.
Circumscribe a hexagon.
Find the area of each hexagon.
The area of the circle is average of the area of the two hexagons.
Method of Exhaustion
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Geometry 11.2 Areas of Circles and Sectors

5. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
Start with a circle:
Divide it into a number of wedges.
As the number of wedges is increased, the shape of each gets closer and closer to an isosceles triangle.
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

6. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
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Geometry 11.2 Areas of Circles and Sectors

7. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
Separate the top half from the bottom half.
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Geometry 11.2 Areas of Circles and Sectors

8. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
Peel the wedges apart and line them up.
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Geometry 11.2 Areas of Circles and Sectors

9. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
Slip them together.
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Geometry 11.2 Areas of Circles and Sectors

10. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
(Allowing for poor artistic rendering….)
This shape looks like a ____________.
parallelogram
The height is the same as the ______ of the circle.
radius
The length of the base is the same as the length of a _________ of the circle.
semicircle r r
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Geometry 11.2 Areas of Circles and Sectors

11. Geometry 11.2 Areas of Circles and Sectors
Method 3: Use Triangles
r2
The area of the parallelogram is A = ____. r r
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Geometry 11.2 Areas of Circles and Sectors

12. Geometry 11.2 Areas of Circles and Sectors
Area of a Circle r
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

13. Geometry 11.2 Areas of Circles and Sectors
Example 1
Find the area of the circle.
Solution:
A = (72) = 49 in2 or
A  in2
7 in
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Geometry 11.2 Areas of Circles and Sectors

14. Geometry 11.2 Areas of Circles and Sectors
Example 2
The area of a circle is 400 ft2. Find the radius of the circle.
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Geometry 11.2 Areas of Circles and Sectors

15. Geometry 11.2 Areas of Circles and Sectors
Example 3
At Moldy Cheese Pizza they sell two sizes of pizza: medium and large. The medium has a diameter of 10 inches and the large has a diameter of 16 inches.
On Fridays they have a special: you get one large pizza, or two medium pizzas for the same price. Which is the better deal?
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

16. Geometry 11.2 Areas of Circles and Sectors
Large
Diameter = 16
Radius = ?
Medium
Diameter = 10
Radius = ? 8 5 5 8
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

17. Geometry 11.2 Areas of Circles and Sectors
Large
Radius = 8
Area = ?
Medium
Radius = 5
Area = ?
64
25 5 8
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Geometry 11.2 Areas of Circles and Sectors

18. Geometry 11.2 Areas of Circles and Sectors
Large
Area = 64
Medium
Area = 25 5 8
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

19. 50 8 5 5 Large Area = 64 Medium Area = 25
So two medium pizzas have an area of
50 8 5 5
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Geometry 11.2 Areas of Circles and Sectors

20. Geometry 11.2 Areas of Circles and Sectors
Solution: Buying one large has more area than two mediums.
64 8
25
25
201 in2 5 5
157 in2
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Geometry 11.2 Areas of Circles and Sectors

21. Geometry 11.2 Areas of Circles and Sectors
Circle Sectors
Circle Sector
A circle sector is the region bounded by two radii and the intercepted arc of a circle.
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

22. Geometry 11.2 Areas of Circles and Sectors
Area of Circle Sectors
The area of a circle sector is proportional to the measure of the central angle.
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Geometry 11.2 Areas of Circles and Sectors

23. Geometry 11.2 Areas of Circles and Sectors
Area of Circle Sectors
The area of a circle sector is proportional to the measure of the central angle. x r
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Geometry 11.2 Areas of Circles and Sectors

24. Geometry 11.2 Areas of Circles and Sectors
Example 4
Find the area of the sector.
50 10
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Geometry 11.2 Areas of Circles and Sectors

25. Geometry 11.2 Areas of Circles and Sectors
Your Turn
Find the area of sector ABC. A
125 C B 4
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Geometry 11.2 Areas of Circles and Sectors

26. Geometry 11.2 Areas of Circles and Sectors
Example 5
The area of a sector of a circle is 32. What is the measure of the central angle if the radius of the circle is 6?
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Geometry 11.2 Areas of Circles and Sectors

27. Geometry 11.2 Areas of Circles and Sectors
Problem Solving
Read the problem.
What do you need to know to solve to?
What information is given that you can use?
Identify the steps needed.
Solve it and answer the question.
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

28. Geometry 11.2 Areas of Circles and Sectors
Problem
To water a square yard, a man installs four sprinkler heads as shown. The sprinkler heads spray water in a circular pattern with a radius of 4 feet. After a while, he notices some the grass is dying. What is the area of the yard that doesn’t get any water?
4 ft
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors

29. Geometry 11.2 Areas of Circles and Sectors
Problem Solution
Area of square
A = 162 = 256 ft2
Area of one circle
A = (42) = 16 ft2
Area of four circles
A = 4(16) = 64
A  201 ft2
4 ft
16 ft ?
8 ft ?
16 ft ?
Dying Area
256 – 201 = 55 ft2
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Geometry 11.2 Areas of Circles and Sectors

30. Geometry 11.2 Areas of Circles and Sectors
Your Turn
Find the area of the (ring) – the shaded region.
5 in
2 in
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Geometry 11.2 Areas of Circles and Sectors

31. Geometry 11.2 Areas of Circles and Sectors
Solution
Find the area of the (ring).
Area of Outer Circle
Radius = 7 in.
Area = 49
Area of Inner Circle
Radius = 5
Area = 25
25
5 in
2 in 7
49
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Geometry 11.2 Areas of Circles and Sectors

32. Geometry 11.2 Areas of Circles and Sectors
Solution
Outer Circle Area = 49
Inner Circle Area = 25
Area
Outer – Inner =
49 - 25 = 24
 75.4 in2
25
5 in
2 in
49
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Geometry 11.2 Areas of Circles and Sectors

33. Geometry 11.2 Areas of Circles and Sectors
Summary
Area of a circle = r2
Area of a sector is proportional to the measure of the central angle (or the measure of the intercepted arc).
Leave answers in terms of pi when possible.
February 17, 2019
Geometry 11.2 Areas of Circles and Sectors