11

Section 2.2 – Arc Length and Sector Area

Arc Length

Definition

If a central angle, in a circle of a radius r, cuts off an arc of length s, then the measure of, in radians is:

rr

r s

s r ( in radians)

Note:

When applying the formula s r , the value of must be in radian.

Example

A central angle in a circle of radius 3 cm cuts off an arc of length 6 cm. What is the radian measure of

.

Solution

sr

63

cmcm

2 rad

12

Example

A circle has radius 18.20 cm. Find the length of the arc intercepted by a

central angle with measure 38 radians.

Solution

Given: 3 , 18.20 8

rad r cm

rs

18.20 38

cm

21.44 cm

Example

The minute hand of a clock is 1.2 cm long. To two significant digits, how far does the tip of the minute

hand move in 20 minutes?

Solution

Given: r = 1.2 cm

One complete rotation = 1 hour = 60 minutes = 2

6020

2

26020

3

2

rs

322.1

cm 5.2

13

Example

A person standing on the earth notices that a 747 jet flying overhead subtends an angle 0.45. If the length

of the jet is 230 ft., find its altitude to the nearest thousand feet.

Solution

rs

sr

180

45.0

230

45.0

)180(230

ft000,29

Example

A rope is being wound around a drum with radius 0.8725 ft. How much rope will be wound around the

drum if the drum is rotated through an angle of 39.72?

Solution

s r

1800.8725 39.72

0.6049 feet

14

Area of a Sector

A sector of a circle is a portion of the interior of a circle intercepted by a central angle.

2

sec

ircle 2

Area of tor Central angle

Area of c One full rotation

A

r

2 2

2 2 A r

rr

21

2A r

Definition

If (in radians) is a central angle in a circle with radius r, then the area of the sector formed by an angle

is given by

212

A r ( in radians)

Example

Find the area of the sector formed by a central angle of 1.4 radians in a circle of radius 2.1 meters

Solution

Given: r = 2.1 m

= 1.4

221 rA

)4.1(2)1.2(21

2 1.3 m

15

Example

If the sector formed by a central angle of 15 has an area of 3 cm

2, find the radius of a circle.

Solution

Given: 12180

15

3A

221 rA

12

221

3 r

2412

221

324 r

28 r

8r

cmr 22

Example

A lawn sprinkler located at the corner of a yard is set to rotate 90 and project water out 30.0 ft. To three

significant digits, what area of lawn is watered by the sprinkler?

Solution

Given: 2

90

ftr 30

221 rA

2

2)30(21

2 707 ft

16

Exercises Section 2.2 – Arc Length and Sector Area

1. The minute hand of a clock is 1.2 cm long. How far does the tip of the minute hand travel in 40

minutes?

2. Find the radian measure if angle, if is a central angle in a circle of radius r = 4 inches, and cuts

off an arc of length s = 12 inches.

3. Give the length of the arc cut off by a central angle of 2 radians in a circle of radius 4.3 inches

4. A space shuttle 200 miles above the earth is orbiting the earth once every 6 hours. How long, in

hours, does it take the space shuttle to travel 8,400 miles? (Assume the radius of the earth is 4,000

miles.) Give both the exact value and an approximate value for your answer.

5. The pendulum on a grandfather clock swings from side to side once every second. If the length of

the pendulum is 4 feet and the angle through which it swings is 20. Find the total distance traveled

in 1 minute by the tip of the pendulum on the grandfather clock.

6. Reno, Nevada is due north of Los Angeles. The latitude of Reno is 40, while that of Los Angeles is

34 N. The radius of Earth is about 4000 mi. Find the north-south distance between the two cities.

7. The first cable railway to make use of the figure-eight drive system was a Sutter Street Railway.

Each drive sheave was 12 feet in diameter. Find the length of cable riding on one of the drive

sheaves.

12 ft

17

8. The diameter of a model of George Ferris’s Ferris wheel is 250 feet, and is the central angle

formed as a rider travels from his or her initial position 0

P to position 1

P . Find the distance

traveled by the rider if = 45 and if =105.

9. Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates

through an angle of 225, through how many degrees will the larger gear rotate?

10. Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates

through an angle of 300, through how many degrees will the larger rotate?

11. The rotation of the smaller wheel causes the larger wheel to rotate. Through how many degrees will

the larger wheel rotate if the smaller one rotates through 60.0?

12. Find the radius of the larger wheel if the smaller wheel rotates 80 when the larger wheel rotates

50.

13. Los Angeles and New York City are approximately 2,500 miles apart on the surface of the earth.

Assuming that the radius of the earth is 4,000 miles, find the radian measure of the central angle

with its vertex at the center of the earth that has Los Angeles on one side and New York City in the

other side.

18

14. Find the number of regular (statute) miles in 1 nautical mile to the nearest hundredth of a mile. (Use

4,000 miles for the radius of the earth).

15. If two ships are 20 nautical miles apart on the ocean, how many statute miles apart are they?

16. If a central angle with its vertex at the center of the earth has a measure of 1, then the arc on the

surface of the earth that is cut off by this angle (knows as the great circle distance) has a measure of

1 nautical mile.

17. How many inches will the weight rise if the pulley is rotated through an angle of 71 50?

Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 6 in?

18. The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are

rotated through 180? Assume the radius of the bicycle wheel is 13.6 in.

19

19. The circular of a Medicine Wheel is 2500 yrs old. There are 27 aboriginal spokes in the wheel, all

equally spaced.

a) Find the measure of each central angle in degrees and in radians.

b) The radius measure of each of the wheel is 76.0 ft, find the circumference.

c) Find the length of each arc intercepted by consecutive pairs of spokes.

d) Find the area of each sector formed by consecutive spokes,

20. Find the radius of the pulley if a rotation of 51.6 raises the weight 11.4 cm.

21. The total arm and blade of a single windshield wiper was 10 in. long and rotated back and forth

through an angle of 95. The shaded region in the figure is the portion of the windshield cleaned by

the 7-in. wiper blade. What is the area of the region cleaned?

20

22. A frequent problem in surveying city lots and rural lands adjacent to curves of highways and

railways is that of finding the area when one or more of the boundary lines is the arc of the circle.

Find the area of the lot.

A B

CD

23. Nautical miles are used by ships and airplanes. They are different from statue miles, which equal

5280 ft. A nautical mile is defined to be the arc length along the equator intercepted by a central

angle AOB of 1 min. If the equatorial radius is 3963 mi, use the arc length formula to approximate

the number of statute miles in 1 nautical mile.

24. The distance to the moon is approximately 238,900 mi. Use the arc length formula to estimate the

diameter d of the moon if angle is measured to be 0.5170.

7

Solution Section 2.2 – Arc Length and Sector Area

Exercise

The minute hand of a clock is 1.2 cm long. How far does the tip of the minute hand travel in 40 minutes?

Solution

min40min 40min 2

60rad

4 .3

rad

s r

4(1.2)3

5.03 cm

Exercise

Find the radian measure if angle , if is a central angle in a circle of radius r = 4 inches, and cuts off

an arc of length s = 12 inches.

Solution

sr

124

3 rad

Exercise

Give the length of the arc cut off by a central angle of 2 radians in a circle of radius 4.3 inches

Solution

Given: 2 , 4.3 rad r in

rs

)2(3.4

in 6.8

8

Exercise

A space shuttle 200 miles above the earth is orbiting the earth once every 6 hours. How long, in hours,

does it take the space shuttle to travel 8,400 miles? (Assume the radius of the earth is 4,000 miles.) Give

both the exact value and an approximate value for your answer.

Solution

sr

84004200

2 rad

2 2 6

rad x hrrad hr

2 (6)

2x

1.91 hrs

Exercise

The pendulum on a grandfather clock swings from side to side once every second. If the length of the

pendulum is 4 feet and the angle through which it swings is 20. Find the total distance traveled in 1

minute by the tip of the pendulum on the grandfather clock.

Solution

Since 20 20•180 9

rad

The length of the pendulum swings in 1 second:

44•

9 9s r ft

.

In 60 seconds, the total distance traveled

460•

9d

80

83.8 feet3

83.8 feet .

9

Exercise

Reno, Nevada is due north of Los Angeles. The latitude of Reno is 40, while that of Los Angeles is 34

N. The radius of Earth is about 4000 mi. Find the north-south distance between the two cities.

Solution

The central angle between two cities: 40 34 6

1806 6

30rad

s r

30

4000

419 miles

Exercise

The first cable railway to make use of the figure-eight drive system was a Sutter Street Railway. Each

drive sheave was 12 feet in diameter. Find the length of cable riding on one of the drive sheaves.

Solution

Since 3

270 270• 180 2

rad

,

The length of the cable riding on one of the drive sheaves is:

s r

3

6•2

9

28.3 feet

10

Exercise

The diameter of a model of George Ferris’s Ferris wheel is 250 feet, and is the central angle formed as a

rider travels from his or her initial position Po to position P1. Find the distance traveled by the rider if =

45 and if =105.

Solution

ftr D 1252

2502

For = 454

rs

4125

ft98

For = 105127

180105

rs

127125

ft230

Exercise

Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates through an

angle of 225, through how many degrees will the larger gear rotate?

Solution

The motion of the larger gear: 180

5225 225 4

rad

The arc length on the smaller gear is:

s r

52.54

9.817477 cm

The arc length on the larger gear is:

s r

9.817477 4.8

9.8174774.8

2.0453

1802 117.0453

11

Exercise

If a central angle with its vertex at the center of the earth has a measure of 1, then the arc on the surface

of the earth that is cut off by this angle (knows as the great circle distance) has a measure of 1 nautical

mile.

Solution

1 11 •60 60 180 10800

rad

s

r

10800 4000

s

400010800

s

1.16 mis

Exercise

If two ships are 20 nautical miles apart on the ocean, how many statute miles apart are they?

Solution

20 1 120 • rad60 3 3 180 540

s

r

540 4000s

4000540

s

23.27s

12

Exercise

Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates through an

angle of 300, through how many degrees will the larger rotate?

Solution

Both gears travel the same arc distance (s), therefore:

1 1 2 2r rs

23.7 300 7.1

180

23.7 3007.1 18

00

18

3.7 3007.1

156

Exercise

The rotation of the smaller wheel causes the larger wheel to rotate. Through how many degrees will the

larger wheel rotate if the smaller one rotates through 60.0?

Solution

Both gears travel the same arc distance (s), therefore:

1 1 2 2r rs

25.23 60.0 8.16

180

25.23 60.08.16 18

80

1 0

5.23 60.08.16

38.5

Exercise

Find the radius of the larger wheel if the smaller wheel rotates 80 when the larger wheel rotates 50.

Solution

1 1 2 2r r

2

11.7 80 50r

2

11.7 80

5018.7 2 r cm

13

Exercise

How many inches will the weight rise if the pulley is rotated through an angle of 71 50?

Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 6 in?

Solution

171 5060 180

s r

19.27 71 5060 180

11.622 in

69.27

sr

rad

6 180 37.08469.27

37 .0846 60

37 5

Exercise

The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated

through 180? Assume the radius of the bicycle wheel is 13.6 in.

Solution

180 rad The distance for the pedal gear:

1 1 24.72 s r in s

For the smaller gear:

22

4.721.38

3.42sr

The wheel distance: 3 2

13.6 3.42 146.12 rs in

14

Exercise

The circular of a Medicine Wheel is 2500 yrs old. There are 27 aboriginal spokes in the wheel, all equally

spaced.

a) Find the measure of each central angle in degrees and in radians.

b) The radius measure of each of the wheel is 76.0 ft, find the circumference.

c) Find the length of each arc intercepted by consecutive pairs of spokes.

d) Find the area of each sector formed by consecutive spokes,

Solution

a) The central angle: 360 4027 3

18

40

0 23 27

rad rad

b) 477.5 2 2 76 ftC r

c) Since 2 17.776 762

7

r r fts

d) Area 21

2r

2 21 76

2 27

2672 ft

Exercise

Find the radius of the pulley if a rotation of 51.6 raises the weight 11.4

cm.

Solution

180

11.4 51

12.7 .6

r cs m

15

Exercise

The total arm and blade of a single windshield wiper was 10 in. long and rotated back and forth through

an angle of 95. The shaded region in the figure is the portion of the windshield cleaned by the 7-in. wiper

blade. What is the area of the region cleaned?

Solution

The total angle: 19951 6

80 3

rad

1A : The area of arm only (not cleaned by the blade).

1

212

1910 736

A

2A : The area of arm and the blade.

2

212

191036

A

The total cleaned area:

2 1A A A

2 21 12 2

19 1910 336 36

82.9 7.46

275.4 in

Exercise

A frequent problem in surveying city lots and rural lands adjacent to curves of highways and railways is

that of finding the area when one or more of the boundary lines is the arc of the circle. Find the area of the

lot.

Solution

Using the Pythagorean theorem:

2 230 40 50AC r

Total area = Area of the sector (ADC) +

Area of the triangle (ABC)

Total area 21 12 180 2

60r AB BC

21 12 180 2

50 60 30 40

21909 yd

A B

CD

16

Exercise

Nautical miles are used by ships and airplanes. They are different from statue miles, which equal 5280 ft.

A nautical mile is defined to be the arc length along the equator intercepted by a central angle AOB of 1

min. If the equatorial radius is 3963 mi, use the arc length formula to approximate the number of statute

miles in 1 nautical mile.

Solution

160 180

1 10800

rad

The arc length: 10800

3963 1 .15s r

There are 1.15 statute miles in 1 nautical mile.

Exercise

The distance to the moon is approximately 238,900 mi. Use the arc length formula to estimate the

diameter d of the moon if angle is measured to be 0.5170.

Solution

s r

238900 0.517180

2156 mi