10.7 Areas of Circles and Sectors

Geometry

Mr. Peebles

Spring 2013

Geometry Bell Ringer

• Find the area of P.

8 in.

P

Geometry Bell Ringer

• Find the area of P. Solution:

a. Use r = 8 in the area formula.

A = r2

= • 82

= 64

201.06

8 in.

P

So, the area if 64, or about 201.06 square inches.

Daily Learning Target (DLT) Friday May 10, 2013

• “I can find the area of a circle and a sector of a circle in real life.”

Assignment Pgs. 570-573

(9-26, 27-39 Odds, 42-47,

52-57, 73-74)

Assignment 9. Arc ED

10. Arc FEB

11. Arc BFE

12. Arcs FE and ED

13. Angle FOE

14. Angles FOE and BOC

15. Arc TC = 128 Degrees

16. Arc TBD = 180 Degrees

Assignment 17. Arc BTC = 218 Degrees

18. Arc TCB = 270 Degrees

19. Arc CD = 52 Degrees

20. Arc CBD = 308 Degrees

21. Arc TCD = 180 Degrees

22. Arc DB = 90 Degrees

23. Arc TDC = 232 Degrees

24. Arc TB = 90 Degrees

Assignment 25. Arc BC = 142

26. Arc BCD = 270 Degrees

27. 20π cm

29. 8.4π m

31. π m 33. 25 in. 35. 8π FT 37. 33π in.

39. 5π/4 m 42. 70 43. 180 44. 110 45. 55 46. 235 47. 290

Assignment 52. 38

53. 40

54. 100 in.

55. 50 in.

56. 100π/3 in.

57. B

73. B

74. G

Examples of regular polygons inscribed in circles.

3 -gon 4 -gon

5 -gon 6 -gon

Theorem. 11.7: Area of a Circle

• The area of a circle is times the square of the radius or A = r2.

r

Ex. 1: Using the Area of a Circle

• Find the diameter of

Z.

Solution:

b. Area of circle Z is 96 cm2.

A = r2

Z

Ex. 1: Using the Area of a Circle

• Find the diameter of

Z.

Solution:

b. Area of circle Z is 96 cm2.

A = r2

96= r2

96= r2

30.56 r2

5.53 r

The diameter of the circle is about 11.06 cm.

Z

More . . .

• A sector of a circle is a region bounded by two radii of the circle and their intercepted arc. In the diagram, sector APB is bounded by AP, BP, and . The following theorem gives a method for finding the area of a sector.

Theorem 11.8: Area of a Sector

• The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360°.

A m

360° = • r2

Ex. 2: Finding the area of a sector

• Find the area of the sector shown below.

4 ft.

P

D

C

Sector CPD intercepts an arc whose measure is 80°. The radius is 4 ft.

A m

360° = • r2

Ex. 2 Solution

A m

360° = • r2

80° 360°

= • 42

11.17

Write the formula for area of a sector.

Substitute known values.

Use a calculator.

So, the area of the sector is about 11.17 square feet.

Ex. 3: Finding the Area of a Sector

• A and B are two points on a P with radius 9 inches and mAPB = 60°. Find the areas of the sectors formed by APB.

9P

B

AQ

60°

FIRST draw a diagram of P and APB. Shade the sectors.

LABEL point Q on the major arc.

FIND the measures of the minor and major arcs.

Ex. 3: Finding the Area of a Sector

• Because mAPB = 60°, m = 60° and

• m = 360° - 60° = 300°.

Use the formula for the area of a sector.

Area of small sector 60° 360°

= • r2

60° 360°

= • • 92

1

6 = • • 81

42.41 square inches

Ex. 3: Finding the Area of a Sector

• Because mAPB = 60°, m = 60° and

• m = 360° - 60° = 300°.

Use the formula for the area of a sector.

Area of large sector 300° 360°

= • r2

60° 360°

= • • 92

5

6 = • • 81

212.06 square inches

Using Areas of Circles and regions

• You may need to divide a figure into different regions to find its area. The regions may be polygons, circles, or sectors. To find the area of the entire figure, add or subtract the areas of separate regions as appropriate.

Ex. 4: Find the Area of a Region

• Find the area of the region shown.

The diagram shows a regular hexagon inscribed in a circle with a radius of 5 meters. The shaded region is the part of the circle that is outside the hexagon.

Area of

Shaded

Area of

Circle

Area of

Hexagon =

Solution: Area of

Shaded

Area of

Circle

Area of

Hexagon =

= r2 ½ aP

= ( • 52 ) – ½ • ( √3) • (6 • 5)

= 25 - √3, or about 13.59

square meters.

75 2

Ex. 5: Finding the Area of a Region

• Woodworking. You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case?

Ex. 5: Finding the Area of a Region

More . . .

• Complicated shapes may involve a number of regions. In example 6, the curved region is a portion of a ring whose edges are formed by concentric circles. Notice that the area of a portion of the ring is the difference of the areas of the two sectors.

Assignment:

• Pgs. 577-580 (1-27 Odds, 41-43)

Geometry Closer

• Find the area of the sector shown below.

4 ft.

P

D

C

Sector CPD intercepts an arc whose measure is 65°. The radius is 4 ft.

A m

360° = • r2

Geometry Closer

A m

360° = • r2

65° 360°

= • 42

907

Write the formula for area of a sector.

Substitute known values.

Use a calculator.

So, the area of the sector is about 9.07 square feet.