2ObjectivesFind the circumference of a circle and the length of a circular arc.Use circumference and arc length to solve real-life problems.
3Finding 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.
4Theorem 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.
5Ex. 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.
6What’s the difference??Find the exact radius of a circle with circumference 54 feet.Find the radius of a circle with circumference 54 feet.
7Extra 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.
8And . . . 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).
9Finding 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• 2rArc length of=360°
10More . . .½ • 2rThe length of a semicircle is half the circumference, and the length of a 90° arc is one quarter of the circumference.d¼ • 2r
11Ex. 2: Finding Arc Lengths Find the length of each arc.a.c.50°100°
12Ex. 2: Finding Arc Lengths Find the length of each arc.b.50°
13Ex. 2: Finding Arc Lengths Find the length of each arc.c.100°
14Ex. 2: Finding Arc Lengths Find the length of each arc.# of °a.• 2ra. Arc length of =360°50°a. Arc length of =50°360°• 2(5) 4.36 centimeters
15Ex. 2: Finding Arc Lengths Find the length of each arc.# of °b.• 2rb. Arc length of =360°50°50°• 2(7)b. Arc length of =360° 6.11 centimeters
16Ex. 2: Finding Arc Lengths Find the length of each arc.# of °c.• 2rc. Arc length of =360°100°100°• 2(7)c. Arc length of =360° centimetersIn 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.
17Ex. 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?