bellwork what is the circumference of a circle with a radius of 10? what is the circumference of a...

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

BellworkWhat is the circumference of a circle with a radius of 10?

14

. 31.4

. 62.8

. 314

A units

B units

C units

BellworkWhat is the area of that same circle?

14

2

2

2

. 31.4

. 314

. 985.96

A units

B units

C units

BellworkHow many degrees are in a circle?

14

. 90

. 180

. 360

A

B

C

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

Circumference and Arc Length

Section 11.4

The Concept Today we’re going to revisit a topic from chapter 1 and then

combine it with what we know from chapter 10

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.

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

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

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

ExampleWhat is the length of the arc show below

15

6036060

2 1536015.7

C

units

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

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

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

On your ownWhat is the length of the blue line

60

100

50

50: 415

: 627

: 715

A units

B units

C units170o

160o

The most famous use of this Erastothenes’ proof of the circumference of the earth

Homework

11.45, 9-12, 19-25

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

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

Bellwork Solution What is the value of a2-2ab+b2, if (a-b)=12

Clickers

.1

.12

.144

.288

A

B

C

D

Areas of Circles and Sectors

Section 11.5

Area of a CircleWe’ve already seen the area of a circle, which is given by the

formulaTheorem 11.9

A=πr2

r

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

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

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

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

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

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

On your ownWhat is the area of the shaded space below

3 m

4 m

Homework

11.52, 15-19, 22, 27-31

Homework

11.4 5, 10-12, 20-24

11.5 2, 16-19, 22, 28-30