Lesson 6.7 Lesson 6.7 Circumference and Arc LengthCircumference 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
Minor ArcMinor Arc•Use 2 lettersUse 2 letters•Angle is less than or equal to 180Angle is less than or equal to 180
XX YY
ZZ
120°120°99
XYZ Major ArcMajor Arc•Use 3 lettersUse 3 letters•Angle is greater than Angle is greater than 180180
XZ
m XZ = m<XCZ = 120o
C
Central Angle: Any angle whose vertex is the center of the circle
Terminology
m XYZ = m<XCZ = 240o
Circumference of a Circle
• The circumference C of a circle is
• C = d or C = 2r, 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
• C = d
• diameter = 14 + 2(5.1) d = 24.2 inches
• circumference = (24.2)
• C ≈ 75.99 inches.
Comparing Circumferences - Tire A
• C = d
• diameter = 15 + 2(5.25)
• d = 25.5 inches
• Circumference = (25.5)
• C ≈ 80.07 inches
Comparing Circumferences - Tire B
• Divide the distance traveled by the tire circumference to find the number of revolutions made.
• First, convert 100 feet to 1200 inches.
TIRE A: 100 ft.75.99 in.
1200 in.75.99 in.= 100 ft.
80.07 in.1200 in.
80.07 in.=
15.8 revolutions
TIRE B:
14.99 revolutions
Comparing CircumferencesTire A vs. Tire B
Revolutions = distance traveled circumference
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.
Portions of a Circle: Determine the Arc measure based on the portion given.
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
90o
90o
½ ● 360
180o
180o
1/3 ● 360
120o
120o
1/6 ● 360
60o
60o
90o 180o
120o
60o
An arc length is a portion of the circumference of a circle.
Arc Length Formula
m˚360˚ 2πrArc Length =
measure of the central angle or arc
The fraction of the circle!
The circumference of the entire 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 length of =360°
• 2r AB
mAB
Arc length linear units (inches/feet/meters …)Arc measure degrees
Arc measure
Finding Arc Lengths
• Find the length of each arc.
5 cm
B
A
50°
a.7 cm
D
C
50°
b. 7 cm
F
E
100°c.
Finding Arc Lengths, con’t.
• Find the length of each arc.
5 cm
B
A
50°
a.
a. Arc length of = AB 50°
360°• 2(5)
a. Arc length of = AB # of °
360°• 2r
4.36 centimetersArc length of
AB
7 cm
D
C
50°
b. b. Arc length of = CD # of °
360°• 2r
b. Arc length of = CD 50°
360°• 2(7)
6.11 centimeters
Finding Arc Lengths, con’t.
• Find the length of each arc.
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.
7 cm
F
E
100°
c.c. Arc length of =
# of °
360°• 2r
c. Arc length of = EF 100°
360°• 2(7)
EF
12.22 centimeters
Finding Arc Lengths, con’t.
• Find the length of each arc.
Arc length of EF
Find the exact length of AB
90
6
m AOB
radius
240
12
m AOB
radius
300
12
m AOB
radius
120
2.4
m AOB
radius
108
10 2
m AOB
radius
Fraction of circle:
90o
90o
6
Fraction ● circumference
¼ ● 12π
3π units
240o
240o
12
Fraction of circle:
Fraction ● circumference
2/3 ● 24π
16π units
300o
300o 12
Fraction of circle:
5/6 ● 24π
20π units
A
AB
B
O OAO
B
120o
120o
2.4AO
B
Fraction of circle:
1/3 ● 4.8π
1.6π units
108o
108o
10√2A
B
O
Fraction of circle:
3/10 ● 20√2π
6√2π units
3
2
360
240
4
1
360
90
6
5
360
300
3
1
360
120
10
3
360
108
3.82 m
60º50º5c
m
dmAB
BarclengthA 360
52360
50 arclength
38.1arclength
d360
6082.3
cmarclength 36.4
d6
182.3
Cdm 92.22
7.64 in.
18 in.
Z
Y
X
Using Arc Lengths• Find the indicated measure.
XY
b. mXY
Arc length of 2r= 360°
18
(15.28)=
135° m
XY
XY
m360° •
XY
Substitute and Solve for XY
m
m
18 in. 2(7.64)= 360°
XY
m
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)
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
• Distance = 2s + 2r1
= 2(108.9) + 2(29.00)
400.0 meters
• Distance = 2s + 2r2
= 2(108.9) + 2(30.25)
407.9 meters
Lane 1
Lane 2
Finding Arc Length, con’t
Find each arc length. Give answers in terms of and rounded to the nearest hundredth.
Finding Arc Length
FG
Use formula for area of sector.
5.96 cm 18.71 cm
Substitute 8 for r and 134 for m.
Simplify.
Find each arc length. Give answers in terms of and rounded to the nearest hundredth.
Finding Arc Length
an arc with measure 62 in a circle with radius 2 m
Use formula for area of sector.
0.69 m 2.16 m
Substitute 2 for r and 62 for 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.
Simplify.= m 4.19 m
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.
= 3 cm 9.42 cm
Substitute 4 for r and 135 for m.
Simplify.
Upcoming
• 6.7 Monday• 6.7 Tuesday• Chapter Review Wednesday• Chapter Review Thursday• Chapter 6 Test Friday