1. 11.1 Circumference and Arc Length 11.2 Areas of Circles and Sectors
You will find arc lengths and other measures.
Essential Questions:
How do you find the length of an arc of a circle?
You will learn how to answer this question by using the ratio of the part of the circumference to the entire circle.
How do you find the area of a sector of a circle?
You will find the areas of circles and sectors.
You will learn how to answer this question by using the ratio of the measure of the intercepted arc to 360 degrees.

2. Use the formula for circumference
EXAMPLE 1
Use the formula for circumference
Find the indicated measures.
a. Circumference of a circle with radius 9 centimeters
SOLUTION a.
C = 2 πr
Write circumference formula.
= 2 π 9
Substitute 9 for r.
= 18 π
Simplify.
56.55
Use a calculator.
The circumference is about centimeters.
ANSWER

3. Use the formula for circumference
EXAMPLE 1
Use the formula for circumference
Find the indicated measures.
b. Radius of a circle with circumference 26 meters
C = 2 πr
Write circumference formula. =
2 πr 26
Substitute 26 for C. = 26 2π r
Divide each side by 2π. r
4.14
Use a calculator.
The radius is about 4.14 meters.
ANSWER

4. EXAMPLE 2
Use circumference to find distance traveled
Tire Revolutions
The dimensions of a car tire are shown at the right. To the nearest foot, how far does the tire travel when it makes 15 revolutions?
SOLUTION
STEP 1
Find the diameter of the tire d
= (5.5)
= 26 in.
STEP 2
Find the circumference of the tire C
= πd
= π(26)
≈ in.

5. EXAMPLE 2
Use circumference to find distance traveled
STEP 3
Find the distance the tire travels in 15 revolutions. In one revolution, the tire travels a distance equal to its circumference. In 15 revolutions, the tire travels a distance equal to 15 times its circumference. 15
81.68 in.
= in.

6. EXAMPLE 2
Use circumference to find distance traveled
STEP 4
Use unit analysis. Change inches to feet.
in.
1 ft
12 in.
= ft
The tire travels approximately 102 feet.
ANSWER

7. GUIDED PRACTICE
for Examples 1 and 2
1. Find the circumference of a circle with diameter 5 inches. Find the diameter of a circle with circumference 17 feet.
about in.; about 5.41 feet
ANSWER

8. GUIDED PRACTICE
for Examples 1 and 2
2. A car tire has a diameter of 28 inches. How many revolutions does the tire make while traveling 500 feet?
about 68 revolutions
ANSWER

9. EXAMPLE 3
Find arc lengths
Find the length of each red arc. a.
SOLUTION
2π(8)
60°
360° = a.
Arc length of AB
≈ 8.38 centimeters

10. EXAMPLE 3
Find arc lengths
Find the length of each red arc. b.
SOLUTION
2π(11)
60°
360° = b.
Arc length of EF
≈ centimeters

11. EXAMPLE 3
Find arc lengths
Find the length of each red arc. c.
SOLUTION
2π(1)
120°
360° = c.
Arc length of GH
≈ centimeters

12. EXAMPLE 4
Find arc lengths to find measures
Find the indicated measure.
a. Circumference C of Z
SOLUTION
Arc length of XY C a.
360°
m XY =
4.19 C
360°
40° =
4.19 C 9 1 =
37.71 = C

13. EXAMPLE 4
Find arc lengths to find measures
Find the indicated measure.
b. m RS
SOLUTION
Arc length of RS
2 r
360° b.
m RS = 44
2 (15.28)
m RS
360° = 44
2 (15.28)
360°
= m RS
165°
m RS

14. GUIDED PRACTICE
for Examples 3 and 4
Find the indicated measure.
3. Length of PQ
about 5.89 yd
ANSWER

15. GUIDED PRACTICE
for Examples 3 and 4
Find the indicated measure.
4. Circumference of N
81.68 m
ANSWER

16. GUIDED PRACTICE
for Examples 3 and 4
Find the indicated measure.
5. Radius of G
about 4.01 ft.
ANSWER

17. EXAMPLE 5
Find arc lengths to find distance
TRACK
The curves at the ends of the track shown are 180° arcs of circles. The radius of the arc for a runner on the red path shown is 36.8 meters. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter.
SOLUTION
The path of a runner is made of two straight sections and two semicircles. To find the total distance, find the sum of the lengths of each part.

18. EXAMPLE 5
Find arc lengths to find distance
= 2(84.39) π 1 2
≈ meters
The runner on the red path travels about 400 meters.
ANSWER

19. GUIDED PRACTICE
for Example 5
6. In Example 5, the radius of the arc for a runner on the blue path is meters, as shown in the diagram. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter.
about meters
ANSWER

20. You will find arc lengths and other measures.
How do you find the length of an arc of a circle?

21. Use the formula for area of a circle
EXAMPLE 1
Use the formula for area of a circle
Find the indicated measure.
SOLUTION
a. Area
r = 2.5 cm
A = πr2
Write formula for the area of a circle.
= π (2.5)2
Substitute 2.5 for r.
= 6.25π
Simplify.
≈ 19.63
Use a calculator.
The area of A is about square centimeters.
ANSWER

22. Use the formula for area of a circle
EXAMPLE 1
Use the formula for area of a circle
Find the indicated measure.
SOLUTION
A = cm2
b. Diameter
A = πr2
Write formula for the area of a circle.
= πr2
Substitute for A.
= r2
113.1 π
Divide each side by π.
≈ r
Find the positive square root of each side.
The radius is about 6 cm, so the diameter is
about 12 centimeters.
ANSWER

23. EXAMPLE 2
Find areas of sectors
Find the areas of the sectors formed by UTV.
SOLUTION
STEP 1
Find the measures of the minor and major arcs.
Because m UTV = 70°, mUV = 70° and
mUSV = 360° – 70° = 290°.

24. Find the areas of the small and large sectors.
EXAMPLE 2
Find areas of sectors
STEP 2
Find the areas of the small and large sectors.
Area of small sector = πr2
mUV
360°
Write formula for area
of a sector.
= π 82
70°
360°
Substitute.
≈ 39.10
Use a calculator.

25. Area of large sector = πr2 mUSV 360°
EXAMPLE 2
Find areas of sectors
Area of large sector = πr2
mUSV
360°
Write formula for area
of a sector.
= π 82
290°
360°
Substitute. ≈
Use a calculator.
The areas of the small and large sectors are about square units and square units, respectively.
ANSWER

26. GUIDED PRACTICE
for Examples 1 and 2
Use the diagram to find the indicated measure.
1. Area of D
about ft2
ANSWER

27. GUIDED PRACTICE
for Examples 1 and 2
Use the diagram to find the indicated measure.
2. Area of red sector
about ft2
ANSWER

28. GUIDED PRACTICE
for Examples 1 and 2
Use the diagram to find the indicated measure.
3. Area of blue sector
about ft2.
ANSWER

29. Use the Area of a Sector Theorem
EXAMPLE 3
Use the Area of a Sector Theorem
Use the diagram to find the area of V.
SOLUTION
Area of sector TVU = Area of V
mTU
360°
Write formula for
area of a sector.
35 = Area of V
40°
360°
Substitute.
315 = Area of V
Solve for Area of V.
The area of V is 315 square meters.
ANSWER

30. EXAMPLE 4
Standardized Test Practice
SOLUTION
The area you need to paint is the area of the rectangle minus the area of the entrance. The entrance can be divided into a semicircle and a square.

31. EXAMPLE 4
Standardized Test Practice
180°
= (26) – (π ) + 162
360°
= 936 – [32π ] ≈
The area is about 579 square feet.
The correct answer is C.
ANSWER

32. GUIDED PRACTICE
for Examples 3 and 4
4. Find the area of H.
about cm2
ANSWER

33. GUIDED PRACTICE
for Examples 3 and 4
5. Find the area of the figure.
about m2
ANSWER

34. GUIDED PRACTICE
for Examples 3 and 4
6. If you know the area and radius of a sector of a circle, can you find the measure of the intercepted arc? Explain.
Yes; the formula for the area of sector is m
A = and if you solve this for m, you get
360
π r2
360A .
ANSWER

35. How do you find the area of a sector of a circle?
You will find the areas of circles and sectors.

36. Daily Homework Quiz
Find the indicated measure.
1. Circumference
ANSWER
about in.

37. Daily Homework Quiz
2. Radius
C = 48 ft
ANSWER
about 7.64 ft

38. Daily Homework Quiz
3. Length of AB
ANSWER
8.64 cm

39. Daily Homework Quiz
4. Find the total circumference of the circles.
ANSWER cm

40. Daily Homework Quiz
Find the indicated measure.
5. Radius
ANSWER
about cm

41. Daily Homework Quiz
6. Circumference
ANSWER
about in.

42. Daily Homework Quiz
1. Find the area of the sectors formed by DEF.
ANSWER
in.2, in.2

43. Daily Homework Quiz
2. Use the area of the sector to find the area
of R
ANSWER
cm2

44. Daily Homework Quiz
3. Find the area of the shaded region.
ANSWER
23.18 ft2