11.4 Circumference and Arc Length

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CIRCLES
Lesson 6.7 Circumference and Arc Length
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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

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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

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Circumference
Distance around the circle
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• Minor Arc
• Use 2 letters
• Angle is less than or equal to 180

Terminology
XZ
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• 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 mltXCZ 120o
m XYZ mltXCZ 240o
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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)

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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
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Comparing Circumferences - Tire A

• C ?d
• diameter 14 2(5.1) d 24.2 inches
• circumference ?(24.2)
• C 75.99 inches.

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Comparing Circumferences - Tire B

• C ?d
• diameter 15 2(5.25)
• d 25.5 inches
• Circumference ?(25.5)
• C 80.07 inches

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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
100 ft.
1200 in.
TIRE B


75.99 in.
75.99 in.
80.07 in.
80.07 in.
? 15.8 revolutions
? 14.99 revolutions
COMPARISON Tire A required more revolutions to
cover the same distance as Tire B.
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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.
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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 6p out of a total 36p on the circle
¼ ? 360
½ ? 360
1/3 ? 360
180o
1/6 ? 360
90o
120o
60o
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Arc Length Formula
measure of the central angle or arc
m
The circumference of the
entire circle!
2pr
Arc Length
360
The fraction of the circle!
.
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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
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Finding Arc Lengths

• Find the length of each arc.

a.
b.
c.
50
50
100
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Finding Arc Lengths, cont.

• Find the length of each arc.

a.
of
2?r
a. Arc length of
360
50
? 4.36 centimeters
Arc length of
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Finding Arc Lengths, cont.

• 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.
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Finding Arc Lengths, cont.

• Find the length of each arc.

c.
of
2?r
c. Arc length of
360
100
100
2?(7)
c. Arc length of
360
? 12.22 centimeters
Arc length of
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Find the exact length of AB
300o
A
120o
108o
240o
B
90o
B
120o
90o
240o
300o
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108o
A
O
A
B
O
A
O
A
O
2.4
O
6
12
10v2
B
B

Fraction of circle
Fraction of circle
Fraction of circle
Fraction of circle
Fraction of circle

Fraction ? circumference
Fraction ? circumference
¼ ? 12p
2/3 ? 24p
5/6 ? 24p
1/3 ? 4.8p
3/10 ? 20v2p
20p units
3p units
16p units
1.6p units
6v2p units
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Find the circumference
Find the arc length
3.82 m
60º
5cm
50º
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Using Arc Lengths

• Find the indicated measure.

m
2?r

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
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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)

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Finding Arc Length, cont

• Find the distance around Lanes 1 and 2.
• The track is made up of
• two semicircles
• two straight sections with length s

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Lane 1
Finding Arc Length, cont

• Distance 2s 2?r1
• 2(108.9) 2?(29.00)
• ? 400.0 meters
• Distance 2s 2?r2
• 2(108.9) 2?(30.25)
• ? 407.9 meters

Lane 2
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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.
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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.
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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.
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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.
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Upcoming

• 6.7 Monday
• 6.7 Tuesday
• Chapter Review Wednesday
• Chapter Review Thursday
• Chapter 6 Test Friday

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