07 Areas of Circles andSectors

Mathematics Florida StandardsMAFS.912.G-C.2.5 Derive., .the formula for the

area of a sector.

MP1. MP 3, MP4, MP6, MP 8

Objective To find the areas of circles, sectors, and segments of circles

Getting Ready! X C

Each of the regular polygons in the table has radius 1. Use a calculatorto complete the table for the perimeter and area of each polygon.Write out the first five decimol places.

Try to find apattern in theseperimeters andareas to tell

you what thecircumference

and area of acircle should be.

MATHEMATICAL

PRACTICES

Look at the results in your table. Notice the perimeter and area of ann-gon as n gets very large. Now consider a circle with radius 1. Whatare the circumference and area of the circle? Explain your reasoning.

Lesson

Vocabularysector of a circle

• segment of acircle

^ In the Solve It, you explored the area of a circle.

Essential Understanding You can find the area of a circle when you know itsI radius. You can use the area of a circle to find the area of part of a circle formed by two

radii and the arc the radii form when they intersect with the circle.

Theorem 10-11 Area of a Circle

The area of a circle is the product of ir and the square of the radius.

A = irr^

II

PolygonNumber

of Sides, n

Lengthof Side, s Apothem, a

Perimeter

{P = ns)

Area

(A = yap)

Decagon 10 2(sin18°) cos 18° 6.18033... 2.93892 ...

ZO-gon 20 2(sln 9°) cos 9° ■ ■

50-gon 50 2(sin 3.6'') cos 3.6° ■ ■

100-gon 100 2(sin 1.8°) cos 1.8° ■ ■

1000-gon 1000 2(sin 0.18°) cos 0.18° ■ ■

660 Chapter 10 Area

What do you need inorder to use the area

formula?

You need the radius. The

diameter is given, soyou can find the radiusby dividing the diameterby 2.

Problem 1 Finding the Area of a Circle

Sports What is the area of the circular regionon the wrestling mat?

Since the diameter of the region is 32 ft, the radiusis Y' or 16 ft

A = 7rr^

= 7t(16)^

= 2567r

Use the area formula.

Substitute 16 forr.

Simplify.

«= 804.2477193 Use a calculator.

The area of the wrestling region is about 804 ft^.

Gotit? 1. a. What is the area of a circular wrestling region with a 42-ft diameter?b. Reasoning If the radius of a circle is halved, how does its

area change? Explain.

What fraction of a

circle's area Is the

area of a sector

formed by a 72° arc?The area of a sector

formed by a 72° arc is

or of the area of

the circle.

A sector of a circle is a region bounded by an arc of the circle and the two

radii to the arc's endpoints. You name a sector using one arc endpoint, thecenter of the circle, and the other arc endpoint.

The area of a sector is a fractional part of the area of a circle. The area of a

sector formed by a 60° arc is or of the area of the circle.

Theorem 10-12 Area of a Sector of a Circle

The area of a sector of a circle is the product of the ratiomeasure of the arc

and the area of the circle360

.

Area of sector AOS =mAB

360Trr

Finding the Area of a Sector of a Circle

What is the area of sector GPH? Leave your answer in terms of tt.

area of sector GPH =inGH

360TTr

360 * """tlS)'= 457r

The area of sector GPH is 45-7r cm^.

Substitute 72 for mGH and 15 for r.

Simplify.

Sector RPS

15 cm

Got It? 2. A circle has a radius of 4 in. What is the area of a sector bounded bya 45° minor arc? Leave your answer in terms of tt.

C PowerGeometry.com Lesson 10-7 Areas of Circles and Sectors 661

A part of a circle bounded by an arc and the segment joining itsendpoints is a segment of a circle.

To find the area of a segment for a minor arc, draw radii to form a sector.

The area of the segment equals the area of the sector minus the area ofthe triangle formed.

Key Concept Area of a Segment

Area of sector Area of triangle = Area of segment

Segmentof a

circle

Problem 3 Finding the Area of a Segment of a Circle

What is the area of the shaded segment shown at the right? Round youranswer to the nearest tenth.

• The radius and m AS

• CA = CB and

m^ACB

The area of sector ACS

and the area of AACS

Subtract the area of

AACS from the area

of sector ACS.

area of sector A CB =mAB

What kind of triangleis AACB7

Since ̂ =CB,the base angles ofAACS are congruent.By the Triangle-Angle-Sum Theorem,mAA = m/LB = 60.

So, AACS is equiangular,and therefore equilateral.

360

360

= 547r

= ̂ •

Trr^ Use the formula for area of a sector.

'7r(18)^ Substitute 60 for mAB and 18 for r.

Simplify.

AACB is equilateral. The altitude forms a 30° -60° -90° triangle,

area of AACB = ̂bh Use the formula for area of a triangle.

18 in, 5= |(18) (9V3) Substitute 18 for b and 9v5 forh.= 8lV3 Simplify,

area of shaded segment = area of sector ACS - area of AACB

= 54ir — 81V3 Substitute.

== 29.34988788 Use a calculator.

The area of the shaded segment is about 29.3 in.^.

Got It? 3. What is the area of the shaded segment shown at the right? Round youranswer to the nearest tenth.

662 Chapter 10 Area

&Lesson Check

Do you know HOW?1. What is the area of a circle with diameter 16 in.?

Leave your answer in terms of tt.

Find the area of the shaded region of the circle. Leave

your answer in terms of tt.

3.

120=

MATHEMATICAL

Do you UNDERSTAND? WSHPRACTICES14. Vocabulary What is the difference between a sector

of a circle and a segment of a circle?

5. Reasoning Suppose a sector of OP has the same

area as a sector of GO. Can you conclude that OP

and 00 have the same area? Explain.

6. Error Analysis Your class

must find the area of a sector

of a circle determined by a

150° arc. The radius of the

circle is 6 cm. What is your

classmate's error? Explain.

y

Practice and Problem-Solving Exercises ̂ ^PRAalcE^^Practice Find the area of each circle. Leave your answer in terms of it. ^ See Problem 1.

7. 9.

1.7ft

11. Agriculture Some farmers use a circular irrigation method. An irrigation arm actsas the radius of an irrigation circle. How much land is covered with an irrigationarm of 300 ft?

12. You use an online store locator to search for a store within a 5-mi radius of yourhome. What is the area of your search region?

Find the area of each shaded sector of a circle. Leave your answer in terms of 97. ^ See Problem 2.

16 cm

16 cm

C PowerGeometry.com Lesson 10-7 Areas of Circles and Sectors 663

Find the area of sector TOP in OO using the given information. Leave youranswer in terms of 77.

19. r = 5m, mTP =90

21. rf = 16m., m^ = 135

20. r = 6 ft, mTP = 15

22. d= 15 cm, mPOT = 180

Find the area of each shaded segment. Round your answer to the nearest tenth. ^ See Problem 3.

23. 120°

^ Apply

Find the area of the shaded region. Leave your answer in terms of tt and in

simplest radical form.

26.

29.-- 4 n

32. Transportation A town provides bus transportation to students living beyond 2 mi

of the high school. What area of the town does not have the bus service? Round tothe nearest tenth.

33. Design A homeowner wants to build a circular patio. If the diameter of the patio is

20 ft, what is its area to the nearest whole number?

34. Think About a Plan A circular mirror is 24 in. wide and has a 4-in. frame around it.

What is the area of the frame?

• How can you draw a diagram to help solve the problem?

• What part of a circle is the width?

• Is there more than one area to consider?

35. Industrial Design Refer to the diagram of the regular hexagonal nut.What is the area of the hexagonal face to the nearest millimeter?

36. Reasoning AB and CD are diameters of © O. Is the area of sector AOCequal to the area of sector BOD? Explain.

37. A circle with radius 12 mm is divided into 20 sectors of equal area. What

is the area of one sector to the nearest tenth?

4 mm

2 mm

mm

664 Chapter 10 Area

38. The circumference of a circle is 267r in. What is its area? Leave your answer in terms

of IT.

39. In a circle, a 90° sector has area SOtt in.^. What is the radius of the circle?

40. Open-Ended Draw a circle and a sector so that the area of the sector is IOtt cm^.Give the radius of the circle and the measure of the sector's arc.

41. A method for finding the area of a segment determined by a minor arc is described

in this lesson.

a. Writing Describe two ways to find the area of a segment determined by amajor arc.

b. If mAB = 90 in a circle of radius 10 in., find the areas of the two segments

determined by AB.

Find the area of the shaded segment to the nearest tenth.

42. 43.

0 Challenge Find the area of the shaded region. Leave your answer in terms of tt.45.

2ft

75°

47.

.-•10 m

48. Recreation An 8 ft-by-lO ft floating dock is anchored in the middle of a pond.The bow of a canoe is tied to a corner of the dock with a lO-ft rope, as shown in thepicture below.

a. Sketch a diagram of the region in which the bow of the canoe can travel.

b. What is the area of that region? Round your answer to the nearest square foot.

'V

C PowerGeometry^com^ Lesson 10-7 Areas of Circles and Sectors 665

49. O O at the right is inscribed in square ABCD and circumscribed A

about square PQRS. Which is smaller, the blue region or the yellowregion? Explain.

50. Circles T and U each have radius 10 and TU = 10. Find the area of the

region that is contained inside both circles. {Hint: Think about where T

and U must lie in a diagram of O Tand O U.) D

:©i Apply What You've Learned

Look back at the information given about the target on page 613. The

diagram of the target is shown again below. In the Apply Wliat You Learned inLesson 10-1, you found tlie area of one red triangle, and in the Apply What

You've Learned in Lesson 10-5, you found the area of the regular octagon.

9 in.

MATHEMATICAL

PRACTICES

MR 1, MR 7

a. Is each yellow region of the target called a segment or a sector of © O?

b. Do the eight yellow regions all have the same area? Justify your answer.

c. What information do you need in order to find the area of the yellow regions of the

target? Describe a method to find this information.

d. Describe a method to find the total area of the yellow regions of the target. Thenfind the total area of the yellow regions. Round your answer the nearest tenth of a

square inch.

e. Use a different method to find the total area of the yellow regions of the target and

check that you get the same result as in part (d).

666 Chapter 10 Area