Circumference and Arc Length CIRCLES Lesson 6.7 Circumference and Arc Length

Objectives/Assignment Find the circumference of a circle and the length of a circular arc. Use circumference and arc length to solve real-life problems. Homework: Lesson 6.7/1-11, 17, 19, 22 Quiz Wednesday Chapter 6 Test Friday

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:  or pi. The exact value of Pi =  The approximate value of Pi ≈ 3.14

Distance around the circle Circumference Distance around the circle

Central Angle: Any angle whose vertex is the center of the circle Z 120° Minor Arc Use 2 letters Angle is less than or equal to 180 Terminology XZ 9 Major Arc Use 3 letters Angle is greater than 180 C XYZ Central Angle: Any angle whose vertex is the center of the circle m XZ = m<XCZ = 120o m XYZ = m<XCZ = 240o

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 (2r = d)

Comparing Circumferences Tire Revolutions Tires from two different automobiles are shown. How many revolutions does each tire make while traveling 100 feet? Tire A Tire B

Comparing Circumferences - Tire A C = d diameter = 14 + 2(5.1) d = 24.2 inches circumference = (24.2) C ≈ 75.99 inches.

Comparing Circumferences - Tire B C = d diameter = 15 + 2(5.25) d = 25.5 inches Circumference = (25.5) C ≈ 80.07 inches

Comparing Circumferences Tire A vs. Tire B Divide the distance traveled by the tire circumference to find the number of revolutions made. First, convert 100 feet to 1200 inches. Revolutions = distance traveled circumference 100 ft. 1200 in. TIRE A: TIRE B: 100 ft. 1200 in. = = 75.99 in. 75.99 in. 80.07 in. 80.07 in.  15.8 revolutions  14.99 revolutions COMPARISON: Tire A required more revolutions to cover the same distance as Tire B.

Arc Length The length of part of the circumference. The length of the arc depends on what two things? 1) The measure of the arc. 2) The size of the circle (radius). An arc length measures distance while the measure of an arc is in degrees.

An arc length is a portion of the circumference of a circle.  Portions of a Circle: Determine the Arc measure based on the portion given. 180o 120o 90o 60o 90o 180o 120o 60o A. B. C. D. ¼ of a circle: ½ of a circle: 1/3 of circumference : 6π out of a total 36π on the circle: ¼ ● 360 ½ ● 360 1/3 ● 360 180o 1/6 ● 360 90o 120o 60o

m˚ 2πr . Arc Length Formula 360˚ measure of the central angle or arc The circumference of the entire circle! 2πr Arc Length = 360˚ The fraction of the circle! .

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 measure m Arc length of = • 2r 360° Arc length  linear units (inches/feet/meters …) Arc measure  degrees

Finding Arc Lengths Find the length of each arc. a. b. c. 50° 50° 100°

Finding Arc Lengths, con’t. Find the length of each arc. a. # of ° • 2r a. Arc length of = 360° 50° a. Arc length of = 50° 360° • 2(5)  4.36 centimeters Arc length of

Finding Arc Lengths, con’t. Find the length of each arc. b. # of ° • 2r b. Arc length of = 360° 50° 50° • 2(7) b. Arc length of = 360°  6.11 centimeters Arc length of In parts (a) and (b), note that the arcs have the same measure but different lengths because the circumferences of the circles are not equal.

Finding Arc Lengths, con’t. Find the length of each arc. c. # of ° • 2r c. Arc length of = 360° 100° 100° • 2(7) c. Arc length of = 360°  12.22 centimeters Arc length of

Find the exact length of AB 12 120o 108o O B O A A O A A 6 12 O 2.4 O 10√2 B B Fraction of circle: Fraction of circle: Fraction of circle: Fraction of circle: Fraction of circle: Fraction ● circumference Fraction ● circumference ¼ ● 12π 2/3 ● 24π 5/6 ● 24π 1/3 ● 4.8π 3/10 ● 20√2π 3π units 16π units 20π units 1.6π units 6√2π units

Find the circumference Find the arc length 3.82 m 5cm 60º 50º

Using Arc Lengths Find the indicated measure. = b. m = = 135°  m m Arc length of 360° b. m Substitute and Solve for m m 18 in. = 2(7.64) 360° 18 = m 360° • (15.28) 135°  m

Finding Arc Length Race Track. The track shown has six lanes. Each lane is 1.25 meters wide. There is 180° arc at the end of each track. The radii for the arcs in the first two lanes are given. Find the distance around Lane 1. (use r1) Find the distance around Lane 2. (use r2)

Finding Arc Length, con’t Find the distance around Lanes 1 and 2. The track is made up of two semicircles two straight sections with length s

Finding Arc Length, con’t Lane 1 Distance = 2s + 2r1 = 2(108.9) + 2(29.00)  400.0 meters Distance = 2s + 2r2 = 2(108.9) + 2(30.25)  407.9 meters Lane 2

Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. FG Use formula for area of sector. Substitute 8 for r and 134 for m.  5.96 cm  18.71 cm Simplify.

Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. an arc with measure 62 in a circle with radius 2 m Use formula for area of sector. Substitute 2 for r and 62 for m.  0.69 m  2.16 m Simplify.

Check It Out! Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth. GH Use formula for area of sector. Substitute 6 for r and 40 for m. =  m  4.19 m Simplify.

Check It Out! Find each arc length. Give your answer in terms of  and rounded to the nearest hundredth. an arc with measure 135° in a circle with radius 4 cm Use formula for area of sector. Substitute 4 for r and 135 for m. = 3 cm  9.42 cm Simplify.

Upcoming 6.7 Monday 6.7 Tuesday Chapter Review Wednesday Chapter Review Thursday Chapter 6 Test Friday