11.4 Circumference and Arc Length

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11.4 Circumference and Arc Length

• Geometry
• Mrs. Spitz
• Spring 2006

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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.
• Assignment pp. 686-688 1-38

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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.
  This ratio is known as ? or pi.

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Theorem 11.6 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.

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Ex. 1 Using circumference

• Find (a) the circumference of a circle with
  radius 6 centimeters and (b) the radius of a
  circle with circumference 31 meters. Round
  decimal answers to two decimal places.

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Solution
b.

• C 2?r
• 2 ? 6
• 12?
• ? 37.70
• ?So, the circumference is about 37.70 cm.

• C 2?r
• 31 2?r
• 31 r
• 4.93 ? r
• ?So, the radius is about 4.93 cm.

a.
2?
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And . . .

• An arc length is a portion of the circumference
  of a circle. You can use the measure of an arc
  (in degrees) to find its length (in linear units).

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Arc Length Corollary

• 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
m

2?r
360
m
or Arc length of

2?r
360
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More . . .

• The length of a semicircle is half the
  circumference, and the length of a 90 arc is one
  quarter of the circumference.

½ 2?r
r
r
¼ 2?r
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Ex. 2 Finding Arc Lengths

• Find the length of each arc.

a.
b.
c.
50
100
50
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Ex. 2 Finding Arc Lengths

• Find the length of each arc.

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

• Find the length of each arc.

of
b.
2?r
b. Arc length of
360
50
50
2?(7)
b. Arc length of
360
? 6.11 centimeters
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Ex. 2 Finding Arc Lengths

• Find the length of each arc.

of
c.
2?r
c. Arc length of
360
100
100
2?(7)
c. Arc length of
360
? 12.22 centimeters
In parts (a) and (b) in Example 2, note that the
arcs have the same measure but different lengths
because the circumferences of the circles are not
equal.
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Ex. 3 Using Arc Lengths

• Find the indicated measure.

a. circumference
Arc length of

2?r
360
3.82
60

2?r
360
60
3.82
1

2?r
6
3.82(6) 2?r
22.92 2?r
C 2?r so using substitution, C 22.92 meters.
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Ex. 3 Using Arc Lengths

• Find the indicated measure.

b. m
Arc length of
m

2?r
360
18
m

360
360
2?(7.64)
360
135 ? m
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Ex. 4 Comparing Circumferences

• Tire Revolutions Tires from two different
  automobiles are shown on the next slide. How
  many revolutions does each tire make while
  traveling 100 feet? Round decimal answers to one
  decimal place.

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Ex. 4 Comparing Circumferences

• Reminder C ?d or 2?r.
• Tire A has a diameter of 14 2(5.1), or 24.2
  inches.
• Its circumference is ?(24.2), or about 76.03
  inches.

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Ex. 4 Comparing Circumferences

• Reminder C ?d or 2?r.
• Tire B has a diameter of 15 2(5.25), or 25.5
  inches.
• Its circumference is ?(25.5), or about 80.11
  inches.

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Ex. 4 Comparing Circumferences

• Divide the distance traveled by the tire
  circumference to find the number of revolutions
  made. First, convert 100 feet to 1200 inches.

100 ft.
1200 in.
TIRE A
100 ft.
1200 in.
TIRE B


76.03 in.
76.03 in.
80.11 in.
80.11 in.
? 15.8 revolutions
? 15.0 revolutions
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Ex. 5 Finding Arc Length

• 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.
• Find the distance around Lane 2.

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Ex. 5 Finding Arc Length

• Find the distance around Lane 1.
• The track is made up of two semicircles and two
  straight sections with length s. To find the
  total distance around each lane, find the sum of
  the lengths of each part. Round decimal answers
  to one decimal place.

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Ex. 5 Lane 1

• 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

Ex. 5 Lane 2
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Upcoming

• 11.5 Monday
• 11.6 Wednesday
• Chapter 11 Test Friday w/review before the test.
• Binder check
• Chapter 12 Postulates/Thms.
• Chapter 12 Definitions