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Bellwork What is the circumference of a circle with a radius of 10? What is the circumference of a circle with a radius of 10? What is the area of that same circle? What is the area of that same circle? How many degrees are in a 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? 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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Bellwork What is the circumference of a circle with a radius of 10? What is the circumference of a circle with a radius of 10? 14

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Bellwork What is the area of that same circle? What is the area of that same circle? 14

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Bellwork How many degrees are in a circle? How many degrees are in a circle? 14

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Bellwork 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? 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

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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 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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Circumference The formula for circumference is given as Theorem 11.8 C=πd where d is diameter C=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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Example This 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

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Central Angles and %’s It’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

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Arc Length We 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 This is the amount of the circle that the arc travels through This is the circumference of circle

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Example What is the length of the arc show below 15 60

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On your Own What is the length of the arc show below 22 75

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On your own What is the length of the arc show below 8 195

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On your own What is the measure of the central angle, θ, in the figure below if the length of the arc is 12π units? 14 θ

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On your own What is the length of the blue line 60 100 50 170 o 160 o

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

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Homework 11.4 5, 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. 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 a 2 -2ab+b 2, if (a-b)=12 What is the value of a 2 -2ab+b 2, 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. 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

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

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Areas of Circles and Sectors Section 11.5

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Area of a Circle We’ve already seen the area of a circle, which is given by the formula Theorem 11.9 A=πr 2 r

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Sectors A 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 This is the amount of the circle that the arc travels through This is the area of circle

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Sectors What is the area of the sector shown below 15 60

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On your Own What is the area of the sector shown below 22 110

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On your own What is the area of the sector shown below 10 200

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On your own Given the arc lengths, what is the measure of the Area of the sector (shown in green) below? 15 31.42 u 18.33 u

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Practical Example Pizza 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?

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On your own What is the area of the shaded space below 3 m 4 m

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Homework 11.5 2, 15-19, 22, 27-31

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Homework 11.4 5, 10-12, 20-24 5, 10-12, 20-2411.5 2, 16-19, 22, 28-30 2, 16-19, 22, 28-30

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Keystone Geometry 1 Area and Circumference of a Circle.

Keystone Geometry 1 Area and Circumference of a Circle.

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