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Section 11.4 Circumference and Arc Length Theorem Theorem 11.8: Circumference of a Circle Circumference is the distance around a circle. C = πd , when using diameter. C = 2πr , when using radius. C= dπ = 2πr
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Use the formula for circumference
EXAMPLE 1 Use the formula for circumference Use circumference formula to find the indicated measures. a. Exact circumference of a circle with radius 9 centimeters. Leave π in your answer. C = 2 πr Write circumference formula. = 2 π 9 Substitute 9 for r. = 18 π Simplify. Circumference is about 18 π cm.
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Use the formula for circumference
EXAMPLE 1 Use the formula for circumference b. Radius of a circle with circumference 26 meters. Approximate to nearest hundredth. C = 2 πr Write circumference formula. = 2 πr 26 Substitute 26 for C. = 26 2π r Divide each side by 2π r 4.14 Use calculator to divide 26 by 6.28. Radius is about 4.14 meters.
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EXAMPLE 2 Use circumference to find distance traveled Tire Revolutions The dimensions of a car tire are shown at the right. To the nearest foot, how far does the tire travel when it makes 15 revolutions? STEP 1 Find the diameter of the tire d = (5.5) = 26 in. STEP 2 Find the circumference of the tire C = πd = 3.14(26) ≈ in.
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EXAMPLE 2 Use circumference to find distance traveled STEP 3 Find the distance traveled in 15 revolutions. One revolution equals its circumference. Distance traveled = revolutions circumference. 15 81.68 in. = in. STEP 4 Convert inches traveled to feet traveled. in. 1 ft 12 in. = ft The tire travels approximately 102 feet.
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GUIDED PRACTICE for Examples 1 and 2 1. Find the exact circumference of a circle with diameter 5 inches. Leave π in your answer. C = d π = 5 π Exact circumference is 5π inches. 2. Find the diameter of a circle with circumference 17 feet. Approximate to the nearest hundredth. C = d π = d π 17 = 17 3.14 d Diameter is about 5.41 feet. d 5.41
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Distance traveled = revolutions circumference.
GUIDED PRACTICE for Examples 1 and 2 3. A car tire has a diameter of 28 inches. How many revolutions does the tire make while traveling 500 feet? Distance traveled = revolutions circumference. C = πd = 3.14(28) ≈ in. Find circumference 500 ft revolutions 87.92 in revolutions 87.92 in 6000 in about 68 revolutions
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Section 11.4 Circumference and Arc Length ARC LENGTH An arc length is a _____________ of a circumference of a circle. You use the measure of the _________ in degrees to find its __________________ in linear units. portion arc length
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Section 11.4 Circumference and Arc Length Corollary Arc Length Corollary The ratio of arc length to circumference is equal to the ratio of arc degree to 360. or
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EXAMPLE 3 Find arc lengths Find the length of the indicated arc. a. Length of Leave π in your answer. 2π(12) 60° 360° = Length of AB = 4π cm
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EXAMPLE 3 Find arc lengths Find the length of each indicated arc. b. Length of Approximate to nearest hundredth. 2(3.14)(11) 120° 360° = Length of GH ≈ cm
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EXAMPLE 4 Find arc lengths to find measures Find the indicated measure. a. Circumference C of Z Arc length of XY C 360° m XY = 4.19 C 360° 40° = 4.19 C 9 1 = 37.71 = C C = in
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EXAMPLE 4 Find arc lengths to find measures Find the indicated measure. b. m RS (to nearest whole degree measure) Arc length of RS 2 r 360° m RS = 44 2(3.14)(15.28) m RS 360° = 44 95.96 360° = m RS 165° m RS
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GUIDED PRACTICE for Examples 3 and 4 Find the indicated measure. 4. Length of PQ 9π yd 75° 360° = Length of PQ ≈ yd
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GUIDED PRACTICE for Examples 3 and 4 Find the indicated measure. 5. Radius of G Arc length of EF 2πr 360° m EF = 10.5 ft 2(3.14)r 360° 150° = 10.5 ft 6.28r 12 5 = 126 ft = 31.4r Radius is about 4.01 ft.
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