11.4 Circumference and Arc Length

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Presentation transcript:

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 = 2r, where d is the diameter of the circle and r is the radius of the circle.

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°. m • 2r Arc length of = 360°

More . . . ½ • 2r The length of a semicircle is half the circumference, and the length of a 90° arc is one quarter of the circumference. d ¼ • 2r

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

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

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

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

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

Ex. 2: Finding Arc Lengths Find the length of each arc. # of ° c. • 2r c. Arc length of = 360° 100° 100° • 2(7) c. Arc length of = 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)