1. Circumference and Arc Length
CIRCLES
Lesson 6.7
Circumference and Arc Length

2. Objectives/Assignment
Find the circumference of a circle and the length of a circular arc.
Use circumference and arc length to solve real-life problems.
Homework:
Lesson 6.7/1-11, 17, 19, 22
Quiz Wednesday
Chapter 6 Test Friday

3. Finding Circumference and Arc Length
The circumference of a circle is the distance around the circle.
For all circles, the ratio of the circumference to the diameter is the same:  or pi.
The exact value of Pi = 
The approximate value of Pi ≈ 3.14

4. Distance around the circle
Circumference
Distance around the circle

5. Central Angle: Any angle whose vertex is the center of the circleZ
120°
Minor Arc
Use 2 letters
Angle is less than or equal to 180
Terminology XZ 9
Major Arc
Use 3 letters
Angle is greater than 180 C
XYZ
Central Angle: Any angle whose vertex is the center of the circle
m XZ = m<XCZ = 120o
m XYZ = m<XCZ = 240o

6. Circumference of a Circle
The circumference C of a circle is
C = d or C = 2r, where
d is the diameter of the circle and
r is the radius of the circle (2r = d)

7. Comparing Circumferences
Tire Revolutions
Tires from two different automobiles are shown.
How many revolutions does each tire make while traveling 100 feet?
Tire A
Tire B

8. Comparing Circumferences - Tire A
C = d
diameter = (5.1) d = 24.2 inches
circumference = (24.2)
C ≈ inches.

9. Comparing Circumferences - Tire B
C = d
diameter = (5.25)
d = 25.5 inches
Circumference = (25.5)
C ≈ inches

10. Comparing Circumferences Tire A vs. Tire B
Divide the distance traveled by the tire circumference to find the number of revolutions made.
First, convert 100 feet to 1200 inches.
Revolutions = distance traveled circumference
100 ft.
1200 in.
TIRE A:
TIRE B:
100 ft.
1200 in. = =
75.99 in.
75.99 in.
80.07 in.
80.07 in.
 15.8 revolutions
 revolutions
COMPARISON: Tire A required more revolutions to cover the same distance as Tire B.

11. Arc Length The length of part of the circumference.
The length of the arc depends on what two things?
1) The measure of the arc.
2) The size of the circle (radius).
An arc length measures distance while
the measure of an arc is in degrees.

12. An arc length is a portion of the circumference of a circle.
 Portions of a Circle: Determine the Arc measure based on the portion given.
180o
120o
90o
60o
90o
180o
120o
60o A. B. C. D.
¼ of a circle:
½ of a circle:
1/3 of circumference :
6π out of a total 36π on the circle:
¼ ● 360
½ ● 360
1/3 ● 360
180o
1/6 ● 360
90o
120o
60o

13. m˚ 2πr . Arc Length Formula 360˚ measure of the central angle or arc
The circumference of the
entire circle!
2πr
Arc Length =
360˚
The fraction of the circle! .

14. Arc Length
In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.
Arc measure m
Arc length of =
• 2r
360°
Arc length  linear units (inches/feet/meters …)
Arc measure  degrees

15. Finding Arc Lengths
Find the length of each arc. a. b. c.
50°
50°
100°

16. Finding Arc Lengths, con’t.
Find the length of each arc. a.
# of °
• 2r
a. Arc length of =
360°
50°
a. Arc length of =
50°
360°
• 2(5)
 4.36 centimeters
Arc length of

17. Finding Arc Lengths, con’t.
Find the length of each arc. b.
# of °
• 2r
b. Arc length of =
360°
50°
50°
• 2(7)
b. Arc length of =
360°
 6.11 centimeters
Arc length of
In parts (a) and (b), note that the arcs have the same measure but different lengths because the circumferences of the circles are not equal.

18. Finding Arc Lengths, con’t.
Find the length of each arc. c.
# of °
• 2r
c. Arc length of =
360°
100°
100°
• 2(7)
c. Arc length of =
360°
 centimeters
Arc length of

19. Find the exact length of AB12
120o
108o O B O A A O A A 6 12 O
2.4 O
10√2 B B
Fraction of circle:
Fraction of circle:
Fraction of circle:
Fraction of circle:
Fraction of circle:
Fraction ● circumference
Fraction ● circumference
¼ ● 12π
2/3 ● 24π
5/6 ● 24π
1/3 ● 4.8π
3/10 ● 20√2π
3π units
16π units
20π units
1.6π units
6√2π units

20. Find the
circumference
Find the
arc length
3.82 m
5cm
60º
50º

21. Using Arc Lengths Find the indicated measure. = b. m = = 135°  m m
Arc length of
360°
b. m
Substitute and Solve for m m
18 in. =
2(7.64)
360° 18 = m
360° •
(15.28)
135°  m

22. Finding Arc Length Race Track. The track shown has six lanes.
Each lane is 1.25 meters wide.
There is 180° arc at the end of each track.
The radii for the arcs in the first two lanes are given.
Find the distance around Lane 1. (use r1)
Find the distance around Lane 2. (use r2)

23. Finding Arc Length, con’t
Find the distance around Lanes 1 and 2.
The track is made up of
two semicircles
two straight sections with length s

24. Finding Arc Length, con’t
Lane 1
Distance = 2s + 2r1
= 2(108.9) + 2(29.00)
 meters
Distance = 2s + 2r2
= 2(108.9) + 2(30.25)
 meters
Lane 2

26. Finding Arc Length
Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. FG
Use formula for area of sector.
Substitute 8 for r and 134 for m.
 5.96 cm  cm
Simplify.

27. Finding Arc Length
Find each arc length. Give answers in terms of  and rounded to the nearest hundredth.
an arc with measure 62 in a circle with radius 2 m
Use formula for area of sector.
Substitute 2 for r and 62 for m.
 0.69 m  2.16 m
Simplify.

28. Check It Out!
Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth. GH
Use formula for area of sector.
Substitute 6 for r and 40 for m.
=  m  4.19 m
Simplify.

29. Check It Out!
Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth.
an arc with measure 135° in a circle with radius 4 cm
Use formula for area of sector.
Substitute 4 for r and 135 for m.
= 3 cm  9.42 cm
Simplify.

30. Upcoming 6.7 Monday 6.7 Tuesday Chapter Review Wednesday
Chapter Review Thursday
Chapter 6 Test Friday