Geometry B Bellwork

1
Geometry B Bellwork

• 1) Find the length of the apothem of a regular
  hexagon given a side length of 18 cm.

2
7-6 Circle and Arcs

• Geometry

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. This ratio is known
  as ? or pi.

4
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.

5
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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Arcs

• Arc- part of a circle
• Semicircle- half of the circle
• Minor arc- smaller than a semicircle
• Major arc- greater than a semicircle

8

• Identify the following.
• Minor arcs
• Semicircles
• Major arcs that contain point A

A
D
P
B
C
9
E

• Find the measures of
• the arcs.
• DE
• EA
• EB
• ADB
• EBD

A
D
44
78
P
B
C
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Geometry B Bellwork

• 2)Find the measures of EB, BDE, ADB.

E
A
D
32
82
P
B
C
11
Arc Length Theorem

• The length of an arc of a circle is the product
  of the ratio, measure of the arc
• 360
• and the circumference of a circle.

r
Arc length of
m
AB

AB
2?r
360
12
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
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 AB
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 CD
360
50
50
2?(7)
b. Arc length of CD
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 EF
360
100
100
2?(7)
c. Arc length of EF
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. 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