11.1 Warmup April 28, 2017 Geometry 11.1 Circumference and Arc Length 1
Geometry 11.1 Circumference and Arc Length mbhaub@mpsaz.org
11.1 Essential Question How can you find the length of a circular arc? April 28, 2017 Geometry 11.1 Circumference and Arc Length 3
Goals Find the circumference of a circle. Find the length of an arc. Solve problems using circumference and arc length. arc April 28, 2017 Geometry 11.1 Circumference and Arc Length 4
Circumference The circumference is the distance around a circle. April 28, 2017 Geometry 11.1 Circumference and Arc Length 5
Pi π d The ratio of the circumference to the diameter is called pi π. Use the Greek letter for pi: C C d April 28, 2017 Geometry 11.1 Circumference and Arc Length 6
More About Pi. 3.14159 The fraction 22/7 is close. 355/113 is closer. The Hebrews thought it was 3. Current computer calculations have pi to over 5 trillion places. April 28, 2017 Geometry 11.1 Circumference and Arc Length 7
Pi to a few decimal places: 3.1415926535897932384626433832795028841971693993751 058209749445923078164062862089986280348253421170679 821480865132823066470938446095505822317253594081284 811174502841027019385211055596446229489549303819644 288109756659334461284756482337867831652712019091456 485669234603486104543266482133936072602491412737245 870066063155881748815209209628292540917153643678925 903600113305305488204665213841469519415116094330572 703657595919530921861173819326117931051185480744623 799627495673518857527248912279381830119491298336733 624406566430860213949463952247371907021798609437027 705392171762931767523846748184676694051320005681271 452635608277857713427577896091736371787214684409012 249534301465495853710507922796892589235420199561121 290219608640344181598136297747713099605187072113499 9999837297804995105973173281609631859502445945534 April 28, 2017 Geometry 11.1 Circumference and Arc Length 8
Pi is Irrational does not repeat. does not terminate. There is NO algebraic technique than can be used to find its value. Computing requires the use of calculus. April 28, 2017 Geometry 11.1 Circumference and Arc Length 9
Circumference You can find the circumference if you know the diameter (d) or the radius (r) of the circle. C C 2 d r April 28, 2017 Geometry 11.1 Circumference and Arc Length 10
Example 1 Find the circumference of a circle with a radius of 8. C 2 (8) 16 50.24 This is the circumference in terms of pi. When computing C in decimal, use 3.14 or the π button on your calculator. April 28, 2017 Geometry 11.1 Circumference and Arc Length 11
Arc Length The ratio of the length of an arc to the circumference of a circle is equal to the ratio of the measure of the arc⁰ to 360. A B Arc Length of AB mab 2r 360 Recall: the measure of the arc is equal to the central angle that intercepts it. April 28, 2017 Geometry 11.1 Circumference and Arc Length 12
Example 2 Find the length of AB. A Write the proportion: L B 80 5 Length of Arc Circumference of circle L 80 2 (5) 360 Central Angle April 28, 2017 Geometry 11.1 Circumference and Arc Length 13
Solve the proportion L 80 2 (5) 360 L 2 10 9 9L 20 L 6.98 B A 80 5 L 20 9 6.98 April 28, 2017 Geometry 11.1 Circumference and Arc Length 14
Your Turn Find the length of AB B x A 25 8 x 25 2 (8) 360 x 25 16 360 360x 400 x x 400 360 3.49 10 9 April 28, 2017 Geometry 11.1 Circumference and Arc Length 15
Example 3 A 25 x B 10 The length of AB is 25 and the radius is 10. Find the mab. 25 x 2 (10) 360 20 x 9000 9000 450 x 20 x 143.2 April 28, 2017 Geometry 11.1 Circumference and Arc Length 16
Semicircle Half of a circle. Its measure is 180. Its length is one-half the circumference. April 28, 2017 Geometry 11.1 Circumference and Arc Length 17
Example 4 An indoor track at a gym has the design below. What is the length of the entire track? 40 ft 100 ft April 28, 2017 Geometry 11.1 Circumference and Arc Length 18
Example 4 Solution The ends make one circle. The diameter is 40 ft. C = (40) 40 ft 100 ft April 28, 2017 Geometry 11.1 Circumference and Arc Length 19
Example 4 Solution Add 200 for the straight segments. Perimeter = 40 + 200 ft Or, 325.66 ft. 40 ft 100 ft April 28, 2017 Geometry 11.1 Circumference and Arc Length 20
Practice 1 The circumference of a circle is 200 ft. Find its radius. C 2 r 200 2 r r 200 100 2 31.83 r April 28, 2017 Geometry 11.1 Circumference and Arc Length 21
Practice 2 The radius of a circle is 16 in and central angle ACB measures 42. Find the length of arc AB. x 42 2 (16) 360 2 (16)(42) x 360 x 11.73 April 28, 2017 Geometry 11.1 Circumference and Arc Length 22
Practice 3 An arc of a circle has a length of 12 cm. Its central angle is 15. What is the radius of the circle? 12 15 2 r 360 2 (15) r 12(360) 30 r 4320 r r 4320 30 45.8 April 28, 2017 Geometry 11.1 Circumference and Arc Length 23
Practice 4 A circle has a diameter of 20 cm. An arc on the circle is 10 cm in length. Find the measure of the central angle. 10 x 2 (10) 360 20 x 3600 x x 3600 20 57.3 April 28, 2017 Geometry 11.1 Circumference and Arc Length 24
Summary The length of an arc on a circle is proportional to the measure of the arc and to the central angle. A semicircle has 180 and it s length is half the circumference. April 28, 2017 Geometry 11.1 Circumference and Arc Length 25
Homework mbhaub@mpsaz.org April 28, 2017 Geometry 11.1 Circumference and Arc Length 26