bellwork what is the circumference of a circle with a radius of 10? what is the circumference of a...
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BellworkWhat is the circumference of a circle with a radius of 10?What is the area of that same circle?How many degrees are in a circle?You have decided that you’d like to own 3/5 of the 15 albums by Mariah Carey. How many albums do you want to have?
14
Clickers
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BellworkWhat is the circumference of a circle with a radius of 10?
14
. 31.4
. 62.8
. 314
A units
B units
C units
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BellworkWhat is the area of that same circle?
14
2
2
2
. 31.4
. 314
. 985.96
A units
B units
C units
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BellworkHow many degrees are in a circle?
14
. 90
. 180
. 360
A
B
C
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BellworkYou have decided that you’d like to own 3/5 of the 15 albums by Mariah Carey. How many albums do you want to have?
14
. 9
. 12
. 15
A
B
C
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Circumference and Arc Length
Section 11.4
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The Concept Today we’re going to revisit a topic from chapter 1 and then
combine it with what we know from chapter 10
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CircumferenceThe formula for circumference is given asTheorem 11.8
C=πd where d is diameterC=2πr where r is radius
r
d
Either formula is fine to work with, although it’s important to determine which dimension you
have before calculating
Remember that the decimal equivalent of pi
is 3.14, but it’s much more efficient to either
use the pi button on your calculator or leave it in
terms of pi.
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ExampleThis bicycle wheel has a radius of 30 cm. The height of the tire
I’m going to put on it is 10 cm. Once it’s on my bike, how far will I have traveled after 30 revolutions?
30
10
30 10 40
2
2 40
251.33
251.33 30
7540
r
C r
C
C
Dis
Dis cm
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Central Angles and %’sIt’s important for us to remember that a central angle is one
whose vertex is at the center of the circle and forms an arc that has the same measure as the angle.
In addition to talking about angles, we can use this central angle to discuss a fraction of a circle. Which is given as
θ
360
Portion
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Arc LengthWe can use our central angle to also discuss the physical length
of an arc, which is different, but related to it’s measure in degrees.
The length of this arc can be found by utilizing the angle that the arc travels through
r
θ 360C
This is the amount of the circle that the
arc travels through
This is the circumference of
circle
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ExampleWhat is the length of the arc show below
15
6036060
2 1536015.7
C
units
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On your OwnWhat is the length of the arc show below
22
75 :28.79
:57.57
:2158.5
A units
B units
C units
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On your ownWhat is the length of the arc show below
8
195 : 27.23
: 44.23
:54.45
A units
B units
C units
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On your ownWhat is the measure of the central angle, θ, in the figure below if
the length of the arc is 12π units?
14
θ : 85.72
: 154.29
:167
o
o
o
A
B
C
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On your ownWhat is the length of the blue line
60
100
50
50: 415
: 627
: 715
A units
B units
C units170o
160o
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The most famous use of this Erastothenes’ proof of the circumference of the earth
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Homework
11.45, 9-12, 19-25
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Bellwork The Great Pyramid in Egypt, built about 2500 BCE, took
approximately 20 years to construct. One estimate of the number of stones—each weighing from two to fifteen tons—is 2.3 million. Assume that the laborers worked 365 days per year for 10 hours a day on average. Estimate the number of stones put in place each hour.
What is the value of a2-2ab+b2, if (a-b)=12
Clickers
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Bellwork Solution The Great Pyramid in Egypt, built about 2500 BCE, took
approximately 20 years to construct. One estimate of the number of stones—each weighing from two to fifteen tons—is 2.3 million. Assume that the laborers worked 365 days per year for 10 hours a day on average. Estimate the number of stones put in place each hour.
Clickers
.2 /
.30 /
.31.5 /
.500 /
A stones hr
B stones hr
C stones hr
D stone day
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Bellwork Solution What is the value of a2-2ab+b2, if (a-b)=12
Clickers
.1
.12
.144
.288
A
B
C
D
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Areas of Circles and Sectors
Section 11.5
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Area of a CircleWe’ve already seen the area of a circle, which is given by the
formulaTheorem 11.9
A=πr2
r
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SectorsA sector is a piece of a circle that has a distinct area that can be
determined in a similar way to that of arc length
r
θ
sec
360tor
circle
A
A
This is the amount of the circle that the
arc travels through
This is the area of circle
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SectorsWhat is the area of the sector shown below
15
60
sec
2sec
2sec
360
6015
360
117.81
tor
circle
tor
tor
A
A
A
A units
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On your OwnWhat is the area of the sector shown below
22
110 2
2
2
:42.24
:464.61
:929.21
A units
B units
C units
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On your ownWhat is the area of the sector shown below
10
200
2
2
2
: 34.91
: 56.31
: 174.53
A units
B units
C units
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On your ownGiven the arc lengths, what is the measure of the Area of the
sector (shown in green) below?
15
31.42 u18.33 u
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Practical ExamplePizza from Domino’s comes as a 16” cut into 8 slices. A Extra
Large pizza from D’Bronx comes as a 30” pizza cut into 12 slices. What is the ratio of the areas of a Domino’s slice to a D’Bronx slice?
8 8 4 3275 75 754
28Dominoes : 8
8
215 75D'Bronx :
12 4
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On your ownWhat is the area of the shaded space below
3 m
4 m
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Homework
11.52, 15-19, 22, 27-31
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Homework
11.4 5, 10-12, 20-24
11.5 2, 16-19, 22, 28-30