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11.4 Circumference and Arc Length 2/17/2011

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Objectives Find the circumference of a circle and the length of a circular arc. Use circumference and arc length to solve real-life problems.

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

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What’s the difference?? Find the exact radius of a circle with circumference 54 feet. Find the radius of a circle with circumference 54 feet.

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Extra Examples Write below previous box Find the exact circumference of a circle with diameter of 15. Find the exact radius of a circle with circumference of 25.

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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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Finding the measure of an Arc Length 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 = 360 ° 2r 2r m

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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 ¼ 2 r d

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Ex. 2: Finding Arc Lengths Find the length of each arc. 50 ° a. 100 ° c.

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b. 50 ° Ex. 2: Finding Arc Lengths Find the length of each arc.

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100 ° c. Ex. 2: Finding Arc Lengths Find the length of each arc.

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Ex. 2: Finding Arc Lengths Find the length of each arc. 50 ° a. a. Arc length of = 50 ° 360 ° 2 (5) a. Arc length of = # of ° 360 ° 2 r 4.36 centimeters

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Ex. 2: Finding Arc Lengths Find the length of each arc. 50 ° b. b. Arc length of = # of ° 360 ° 2 r b. Arc length of = 50 ° 360 ° 2 (7) 6.11 centimeters

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Ex. 2: Finding Arc Lengths Find the length of each arc. 100 ° c. c. Arc length of = # of ° 360 ° 2 r c. Arc length of = 100 ° 360 ° 2 (7) 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: Tire Revolutions The dimensions of a car tire are shown. To the nearest foot, how far does the tire travel when it makes 8 revolutions?

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P. 214 (1-10) Assignment

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