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11.4 Circumference and Arc Length 2/17/2011 Objectives Find the circumference of a circle and the length of a circular arc. Use circumference and arc length to solve real-life problems. 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. Theorem 11.6: Circumference of a Circle The circumference C of a circle is C = d or C = 2r, where d is the diameter of the circle and r is the radius of the circle. diameter d 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. 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. 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. 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). 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°. A P B Arc length of AB = m AB 360° • 2r More . . . The length of a semicircle is half the circumference, and the length of a 90° arc is one quarter of the circumference. ½ • 2r d r ¼ • 2r Ex. 2: Finding Arc Lengths Find the length of each arc. E a. 5 cm A 50° c. 7 cm 100° B F Ex. 2: Finding Arc Lengths Find the length of each arc. b. 7 cm C 50° D Ex. 2: Finding Arc Lengths Find the length of each arc. E c. 7 cm 100° F Ex. 2: Finding Arc Lengths Find the length of each arc. a. a. Arc length of AB 5 cm # of ° = A 50° B a. Arc length of AB 360° 50° = • 2r • 2(5) 360° 4.36 centimeters Ex. 2: Finding Arc Lengths Find the length of each arc. b. b. Arc length of CD 7 cm C 50° b. Arc length of CD D # of ° = 360° 50° = • 2r • 2(7) 360° 6.11 centimeters Ex. 2: Finding Arc Lengths Find the length of each arc. E c. c. Arc length of EF 7 cm 100° c. Arc length of EF F # of ° = 360° 100° = • 2r • 2(7) 360° 12.22 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. 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? Assignment P. 214 (1-10)