10-6 Circles and Arcs

10-6

1. Plan

Objectives

1 To find the measures of central angles and arcs

2 To find circumference and arc length

Examples

1 Real-World Connection 2 Identifying Arcs 3 Finding the Measures of Arcs 4 Real-World Connection 5 Finding Arc Length

Math Background

The ratio p of a circle's circumference to its diameter is independent of the size of the circle (C = pd). Ancient calculations of p range from the rather crude 3 to a remarkably accurate 315153. In 1999, a computer calculated the constant p to 206,158,430,000 decimal places. Results such as this are used to check the accuracy of other computer programs.

More Math Background: p. 530D

Lesson Planning and Resources

See p. 530E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to: Finding Circumference Lesson 1-9: Example 2 Extra Skills, Word Problems, Proof

Practice, Ch. 1 Finding Percentages of a Number Skills Handbook, p. 761

566

10-6

Circles and Arcs

What You'll Learn

? To find the measures of

central angles and arcs

? To find circumference and

arc length

. . . And Why

To use the turning radius of a car to compare the distances that its tires travel, as in Example 4

Check Skills You'll Need GO for Help Lesson 1-9 and Skills Handbook, p. 761

Find the diameter or radius of each circle.

1. r = 7 cm, d = 7 14 cm

2. r = 1.6 m, d = 7 3.2 m

3. d = 10 ft, r = 7 5 ft

4. d = 5 in., r = 7 2.5 in.

Round to the nearest whole number.

5. 9% of 360 32

6. 38% of 360 137 7. 50% of 360 180 8. 21% of 360 76

New Vocabulary

? circle ? center ? radius ? congruent circles ? diameter ? central angle ? semicircle ? minor arc ? major arc ? adjacent arcs ? circumference ? pi ? concentric circles ? arc length ? congruent arcs

1 Central Angles and Arcs

Vocabulary Tip

Diameter comes from the classical Greek words dia, meaning through, and meter, meaning measure.

In a plane, a circle is the set of all points equidistant from

C

a given point called the center. You name a circle by its

center. Circle P (P) is shown at the right. A radius is a segment that has one endpoint at the

A

P

B

center and the other endpoint on the circle. PC is a

radius. PA and PB are also radii. Congruent circles

have congruent radii.

A diameter is a segment that contains the center of a circle and has both endpoints on the circle. AB is a diameter.

A central angle is an angle whose vertex is the center of the circle. &CPA is a central angle.

566

64 . 8

. ./ ./ .

000 1111 2222 3333 4444 5555 6666 7777 8888 9999

Chapter 10 Area

1 EXAMPLE Real-World Connection

Gridded Response To learn how people really spend their time, a research firm studied the hour-by-hour activities of 3600 people. The participants were between 18 and 90 years old. Each participant was sent a 24-hour recording sheet every March for three years from 2000 to 2002.

Some information from the study is shown in this circle graph. What is the measure, in degrees, of the central angle used for the Entertainment part?

Sleep Other

15%

AODMNEIT

31% Entertainment

9% Food

20%

7% 18% Must Do

Work

There are 360 degrees in a circle. To find the measure of a central angle in the circle graph, find the corresponding percent of 360. Entertainment is 18%, and 18% of 360 = 0.18 ? 360, or 64.8.

Special Needs L1 Make sure students understand that p is a constant and not a variable. Also, show them how they can estimate the circumference of a circle by approximating 2pr or pd with 6 r or 3 d.

learning style: verbal

Below Level L2 Students may use objects such as aluminum cans and pieces of string to help them understand the formula for the circumference of a circle.

learning style: tactile

You can use the same method to find the measures of the other central angles.

Sleep: 31% of 360 = 111.6

Other:

15% of 360 = 54

Food: 9% of 360 = 32.4

Must Do:

7% of 360 = 25.2

Work: 20% of 360 = 72

Quick Check

E

D

C

1 A

B

B 2 A

3 A

B

B 4 A

5 A

B

E

D

C

D

E

C

D E

C

E

D

C

E

D

C

B

Test-Taking Tip

You can also find the

measure a of a central

angle by using a

proportion. For

Entertainment (18%) in

Example

1:

18 100

5

36a0.

1 a. Critical Thinking Each section of the circle graph represents a measurable

quantity. What is that quantity? number of hours spent doing an activity

b. Each section of the circle graph represents an average. Explain. Each section represents the average of the 3000? participants' answers.

An arc is a part of a circle. One type of arc, a semicircle, is half of a circle. A minor arc is smaller than a semicircle. A major arc is greater than a semicircle.

R

S

T

P

R

S

T

P

R

S

T

P

TRS is a semicircle. mTRS 180

RS is a minor arc. mRS mRPS

RTS is a major arc. mRTS 360 mRS

The measure of a semicircle is 180. The measure of a minor arc is the measure of its corresponding central angle. The measure of a major arc is 360 minus the measure of its related minor arc.

2 EXAMPLE Identifying Arcs

The water line separates a circle into a major arc and a minor arc.

Identify the following in O.

a. the minor arcs 000 0 AD , CE , AC, and DE are minor arcs.

b. the semicircles 111 1 ACE , CED, EDA , and DAC are semicircles.

A

C

O

D

E

c. the major arcs that contain point A 111 1 ACD, CEA, EDC, and DAE are major arcs that contain point A.

Quick Check 2 Identify the four major arcs of O that contain point E. 11 11 CEA , DAE , ACD , EDC

Adjacent arcs are arcs of the same circle that have exactly one point in common. You can add the measures of adjacent arcs just as you can add the measures of adjacent angles.

Key Concepts

Postulate 10-1 Arc Addition Postulate

The measure of the arc formed by two adjacent

B

C

arcs is the sum of the measures of the two arcs.

1 00 mABC = mAB + mBC

A

2. Teach

Guided Instruction

1 EXAMPLE Careers

Statisticians are applied mathematicians. Most public and private companies hire statisticians to gather and analyze data using mathematical techniques. Colleges offer programs to prepare students for careers as statisticians.

2 EXAMPLE

Because two points name two arcs on a circle, naming an arc using just two points can cause confusion. Point out that this book uses two points to name minor arcs and three points to name semicircles and major arcs.

Teaching Tip When you introduce adjacent arcs, ask: If two arcs are adjacent, are their corresponding central angles adjacent? yes What do adjacent angles have in common? one side

Lesson 10-6 Circles and Arcs 567

Advanced Learners L4 After Example 4, ask students to calculate which is greater, the height or the circumference of a can of three tennis balls.

learning style: verbal

English Language Learners ELL Some students may confuse the term circumference with the term circumscribe. Emphasize that circumference is "the length around a circle" and circumscribe is a verb meaning "to draw around."

learning style: verbal

567

3 EXAMPLE Math Tip

Relate the Arc Addition Postulate to the Angle Addition Postulate in Lesson 1-6.

PowerPoint

Additional Examples

1 A researcher surveyed 2000 members of a club to find their ages. The graph shows the survey results. Find the measure of each central angle in the circle graph.

Members' Ages

25% 8%

40%

27%

65 4564

2544 Under 25

65?: 90; 45?64: 144; 25?44: 97.2; Under 25: 28.8

2 Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint.

D A

P

B E mmseiamnjooicrriaarcrrcclesss::1 0 :1 AAADDDEB,0 A,1 ,1 AEAEE;DB ; 3 Find m0 XY and m1 DXM in C.

M

Y

40? D

56? C

W

X m0 XY 96; m1 DXM 236

3 EXAMPLE Finding the Measures of Arcs

Quick Check

Find the measure of each arc.

0

0

a. BC

mBC = m&BOC = 32

0

00 0

b. BD

m0 BD = mBC + mCD

mBD = 32 + 58 = 90

11

c. ABC

A1 BC is a semicircle.

mABC = 180

0

0

d. AB

mAB = 180 - 32 = 148

C B 32

58 D

O

A

3

10 Find m&COD, mCDA, mAD

1 and mBAD.

58; 180; 122; 270

12 Circumference and Arc Length

The circumference of a circle is the distance around the circle. The number pi (p) is the ratio of the circumference of a circle to its diameter.

Key Concepts

Theorem 10-9 Circumference of a Circle The circumference of a circle is p times the diameter.

C = pd or C = 2pr

d

r O

C

Since the number p is irrational, you cannot write it as a terminating or repeating decimal. To approximate p, you can use 3.14, 272, or the key on your calculator.

Circles that lie in the same plane and have the same center are concentric circles.

4 EXAMPLE Real-World Connection

16.1 ft 4.7 ft

Automobiles A car has a turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. In completing the (outer) turning circle, how much farther does a tire travel than a tire on the concentric inner circle?

To find the radius of the inner circle, subtract 4.7 ft from the turning radius.

circumference of outer circle = C = 2pr = 2p(16.1) = 32.2p radius of the inner circle = 16.1 - 4.7 = 11.4 circumference of inner circle = C = 2pr = 2p(11.4) = 22.8p

The difference in the two distances is 32.2p - 22.8p, or 9.4p.

9.4p < 2 9 . 5 3 0 9 7 1 Use a calculator.

A tire on the turning circle travels about 29.5 ft farther than a tire on the inner circle.

Quick Check 4 The diameter of a bicycle wheel is 22 in. To the nearest whole number, how many

revolutions does the wheel make when the bicycle travels 100 ft? 17 revolutions

568 Chapter 10 Area

568

Key Concepts

The measure of an arc is in degrees while the arc length is a fraction of a circle's

circumference.

An

arc

of

608

represents

60 360

or

1 6

of

the

circle.

Its

arc

length

is

1 6

the

circumference of the circle. This observation suggests the following theorem.

Theorem 10-10 Arc Length

The length of an arc of a circle is the product of the ratio

measure of the arc 360

length

and the circumference

0 of AB

0

=

mAB 360

?

2pr

of

the

circle.

A r

O B

5 EXAMPLE Finding Arc Length

Quick Check

Find the length of each arc shown in red. Leave your answer in terms of p.

a.

X

b.

X

O

Y

16 in.

0

le0 ngth of XY

0 length of XY

=

mXY 360

?

pd

=

90 360

?

p(16)

= 4p in.

15 cm

P

O

240

Y 1

le1 ngth of XPY

1 length of XPY

= =

mXPY 360

?

2pr

240 360

?

2p(15)

= 20p cm

5 Find the length of a semicircle with radius 1.3 m. Leave your answer in terms of p. 1.3 m

60

60 It is possible for two arcs of different circles to have the same measure but different

lengths, as shown at the left. It is also possible for two arcs of different circles to

have the same length but different measures. Congruent arcs are arcs that have

the same measure and are in the same circle or in congruent circles.

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

GO

for Help

Example 1 (page 566)

Trash The graph shows types of trash in a typical American city. Find the measure of each central angle to the nearest whole number.

1. Glass 18 2. Metals 29

3. Plastics 40 4. Wood 22

Yard Waste 12%

Food Waste 12%

Other 11%

Paper and Paperboard

35%

43 5. Food Waste

7. Other 40

6. Yard Waste 43

8. Paper and Paperboard 126

Glass

5%

Wood

Plastics 11% Metals 8% 6%

SOURCE: Environmental Protection Agency, 2003. Go to for a data update.

Web Code: aug-9041

Lesson 10-6 Circles and Arcs 569

Guided Instruction

4 EXAMPLE

Connection to Engineering

Ask: If the left wheels travel a different distance from the right wheels in the same amount of time, what can you conclude about their speeds? The left wheels spin faster than the right wheels. The car's differential enables the wheels to do this.

Connection to Algebra

Point out that, although p is a symbol, it is a constant, not a variable.

Error Prevention!

Some students may confuse arc length with the measure of an arc. Point out that arc length is often given in terms of p, unlike the measure of an arc.

PowerPoint

Additional Examples

4 A circular swimming pool with a 16-ft diameter will be enclosed in a circular fence 4 ft from the pool. What length of fencing material is needed? Round to the nearest whole number. 75 ft

1 5 Find the length of A D B in M in terms of p.

B

150? 18 cm M

A

D

21 cm

Resources

? Daily Notetaking Guide 10-6

L3

? Daily Notetaking Guide 10-6--

Adapted Instruction

L1

Closure

One section of a circle graph with a radius of 15 in. is labeled "Radio: 20%." Find the measure and length of the arc corresponding to this section of the circle graph. measure: 72;

569

3. Practice

Assignment Guide

1 A B 1-26, 40-53

2 A B 27-39, 54-69

C Challenge

70-72

Test Prep Mixed Review

73-75 76-83

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 8, 37, 59, 62, 67.

Exercise 14 Have students explain why the two angles are congruent.

Error Prevention!

Exercises 15?26 Remind students that this textbook names minor arcs with two points and semicircles and major arcs with three points.

Exercise 40 Challenge students to construct a central angle on A and construct a congruent central angle on B.

GPS Guided Problem Solving

L3

Enrichment

Reteaching

Adapted Practice

PraNacmetice

Practice 10-6

Find the volume of each pyramid.

1.

54 cm

2.

54 cm

Class 13 in.

45 cm

4.

36 yd 400 yd2

10 in. 5.

10 in. 150 m2

3 m

L4

L2

L1

Date

L3

Volumes of Pyramids and Cones

3. 32 in.

32 in.

34 in.

6. 18 cm

8 cm2

Find the volume of each cone. Round your answers to the nearest tenth.

7.

8.

9.

24 cm 10 cm

12 in. 10 in.

26 m

10.

11.

12.

8 in.

13 in.

15 m

17 m

28 m

2 ft 6 ft

? Pearson Education, Inc. All rights reserved.

Algebra Find the value of the variable in each figure.

13.

14. x

x

15 15 Volume 1500

6 Volume 8

15. 14

9

x

Volume 126

Example 2 (page 567)

Example 3 (page 568)

Example 4 (page 568)

9?14. Answers may vary.

Identify the following in O. Samples are given.

0

1

F

9. a minor arc ED 1

11. a semicircle BFE

10. a major arc FEB

BO

12. a pair of adjac0 ent arcs 0

E

FE and ED 13. an acute central angle 14. a pair of congruent angles C

D

lFOE

lFOE and lBOC

Find the measure of each arc in P.

0

1

1

1

15. TC 128 16. TBD 180 17. BTC 218 18. TCB 270

0

1

1

0

19. CD 52 20. CBD 308 21. TCD 180 22. DB 90

1

0

0

1

23. TDC 232 24. TB 90 25. BC 142 26. BCD 270

C T 128

PD

B

Find the circumference of each circle. Leave your answer in terms of .

27.

20 cm 28.

O

20 cm

6 ft 3 ft

29. 4.2 m

8.4 m

30.

14 in. 31.

m

14 in.

1 2

m

32. 29 cm

58 cm

Example 5 (page 569)

B Apply Your Skills

GO nline

Homework Help

Visit: Web Code: aue-1006

33. The wheel of an adult's bicycle has diameter 26 in. The wheel of a child's bicycle has diameter 18 in. To the nearest inch, how much farther does the larger bicycle wheel travel in one revolution than the smaller bicycle wheel? 25 in.

Find the length of each arc shown in red. Leave your answer in terms of .

34.

35.

14 cm 45

24 ft

7 2

cm

36. 60 8 ft

18 m 27 m

37.

30 36 in.

38. 33 in.

23 m

39.

23 2

m

9 m O

25

5 4

m

40. Use a co0 mpass to draw 0 A and B with diffe0 rent radii.0 Then u0 se a pro0 tractor to draw XY on A and ZW on B so that mXY = mZW. Is XY > ZW? See margin, p. 571.

41. Surveys Use the data in the table to construct a circle graph.

See margin,

Interest in Languages by Students

p. 571.

at McClellan High School

German Japanese Chinese French Spanish

24% 13% 12% 25% 26%

570 Chapter 10 Area

570

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