Lesson 6.7
Circumference and Arc Length
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
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
Distance around the circle
Z
120°
Minor Arc
9
Major Arc
C
X
Y
Central Angle:
XZ
•Use 2 letters
•Angle is less than or equal to 180
•Use 3 letters
•Angle is greater than 180
XYZ
Any angle whose vertex is the center of the circle
m XZ = m<XCZ = 120o
m XYZ = m<XCZ = 240o
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)
diameter d
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
Comparing Circumferences - Tire A
• C = d
• diameter = 14 + 2(5.1)
d = 24.2 inches
• circumference = (24.2)
• C ≈ 75.99 inches.
Comparing Circumferences - Tire B
•
•
•
•
•
C = d
diameter = 15 + 2(5.25)
d = 25.5 inches
Circumference = (25.5)
C ≈ 80.07 inches
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
TIRE A:
100 ft.
75.99 in.
1200 in.
= 75.99 in.
 15.8 revolutions
TIRE B:
100 ft.
80.07 in.
1200 in.
= 80.07 in.
 14.99 revolutions
COMPARISON: Tire A required more revolutions
to cover the same distance as Tire B.
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.
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:
¼ ● 360
½ of a circle:
½ ● 360
1/3 of circumference :
6π out of a total 36π
on the circle:
90o
180o
1/3 ● 360
120o
1/6 ● 360
60o
Arc Length Formula
measure of the central angle or arc
Arc Length =
m˚
The circumference of the
entire circle!
2πr
360˚
The fraction of the circle!
.
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
Arc length of

AB

=
m AB
• 2r
360°
Arc length  linear units (inches/feet/meters …)
Arc measure  degrees
Finding Arc Lengths
• Find the length of each arc.
E
a.
5 cm
A
b.
7 cm
C
50°
50°
c.
7 cm
100°
B
D
F
Finding Arc Lengths, con’t.
• Find the length of each arc.

a.
a. Arc length of AB
5 cm
# of °
=
A
50°
B
• 2r
360°

  4.36 centimeters
50°
a. Arc length of AB
Arc length of AB
=
360°
• 2(5)
Finding Arc Lengths, con’t.
• Find the length of each arc.
b.


b. Arc length of CD
7 cm
C
50°
b. Arc length of CD
D
# of °
=
360°
50°
=
• 2r
• 2(7)
360°
  6.11 centimeters
Arc length of CD
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.
Finding Arc Lengths, con’t.
• Find the length of each arc.
c.
E


  12.22 centimeters
c. Arc length of EF
7 cm
100°
c. Arc length of EF
F
Arc length of EF
# of °
=
360°
100°
=
• 2r
360°
• 2(7)
Find the exact length of AB
A
90o
O
90o
6
300o
240o
240o
B
O
B
300o 12
12
A
A
O
120o
108o
B
120o
O
2.4
A
108o
O 10√2
A
B
B
mAOB  90
mAOB  240
mAOB  300
mAOB  120
mAOB  108
radius  6
radius  12
radius  12
radius  2.4
radius  10 2
Fraction of circle:
Fraction of circle:
Fraction of circle:
108 3

360 10
Fraction of circle:
90 1

360 4
Fraction ● circumference
¼ ● 12π
3π units
Fraction of circle:
240 2

360 3
Fraction ● circumference
2/3 ● 24π
16π units
300 5

360 6
5/6 ● 24π
20π units
120 1

360 3
1/3 ● 4.8π
1.6π units
3/10 ● 20√2π
6√2π units
arclengthAB 
mAB
d

360
60º
50º
50
arclength 
2 5
360
arclength  1.38
arclength  4.36cm
60
3.82 
d
360
1
3.82  d
6
22.92m  d  C
Using Arc Lengths
• Find the indicated measure.

=
Arc length of


m XY
XY
b. m XY
360°
2r

Substitute and Solve for m XY
X

18 in.
m XY
18 in.
=
Z
360° •
7.64 in.
Y
2(7.64)
360°
18
=
(15.28)

m XY

135°  m XY
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.
a. Find the distance around Lane 1. (use r1)
b. Find the distance around Lane 2. (use r2)
Finding Arc Length, con’t
a.

Find the distance around Lanes 1 and 2.
The track is made up of


two semicircles
two straight sections with length s
Finding Arc Length, con’t
Lane 1
• Distance = 2s + 2r1
= 2(108.9) + 2(29.00)
 400.0 meters
Lane 2
• Distance = 2s + 2r2
= 2(108.9) + 2(30.25)
 407.9 meters
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  18.71 cm Simplify.
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.
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.
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.
Upcoming
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•
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6.7 Monday
6.7 Tuesday
Chapter Review Wednesday
Chapter Review Thursday
Chapter 6 Test Friday