Circumference and Arc Length

1. Circumference and Arc Length Wednesday, September 17, 2014 Lesson 6.7 How do we find the circumference of a circle and the length of an arc of a circle? M2 Unit 3: Day 9

2. Warm Ups • Find w, y, and z. Give the answers in • simplest radical form.

3. 2. RT Warm Ups Warm Up Find each measure. 54 3. UQ

4. Circumference The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter. Pi is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to Pi.

5. Helpful Hint The key gives the best possible approximation for on your calculator. Always wait until the last step to round.

6. a. C=2 πr =2 π9 56.55 ANSWER The circumference is about 56.55centimeters. Use the formula for circumference EXAMPLE Find the indicated measures. a. Circumference of a circle with radius 9 centimeters SOLUTION Write circumference formula. Substitute 9 forr. =18 Simplify. Use a calculator.

7. 26 2 πr = 26 r = 2π 4.14 r ANSWER The radius is about 4.14meters. EXAMPLE Use the formula for circumference Find the indicated measures. b. Radius of a circle with circumference 26 meters C =2 πr Write circumference formula. Substitute 26 for c. Divide each side by 2π. Use a calculator.

8. C=d =5 15.71 GUIDED PRACTICE Guided Practice 1. Find the circumference of a circle with diameter 5 inches. Find the diameter of a circle with circumference 17 feet. SOLUTION Write circumference formula. Substitute 5 for d. Use a calculator. The circumference is about 15.71in.

9. C=d = d 17 17 d = 5.41 r GUIDED PRACTICE 1. Find the circumference of a circle with diameter 5 inches. Find the diameter of a circle with circumference 17 feet. circumference = 17 feet Write circumference formula. Substitute 17 for c. Divide each side by π. Use a calculator. The Diameter is about 5.41feet.

10. The arc length of a circle is a fraction of the circumference of the circle.

11. a. 60° a. = 2π(8) Arc length of AB 360° EXAMPLE 3 EXAMPLE Find arc lengths Find the length of each red arc. SOLUTION ≈ 8.38 centimeters

12. b. 60° = b. Arc length of EF 2π(11) 360° EXAMPLE 3 EXAMPLE Find arc lengths Find the length of each red arc. SOLUTION ≈ 11.52 centimeters

13. 120° c. 2π(1) Arc length of GH = 360° c. EXAMPLE 3 EXAMPLE Find arc lengths Find the length of each red arc. SOLUTION ≈ 23.04 centimeters

14. d. Circumference C of Z Arc length of XY d. m XY = 360° C 4.19 40° = 360° C 4.19 1 = 19 C EXAMPLE 4 EXAMPLE Find arc lengths to find measures Find the indicated measure. SOLUTION 37.71 = C

15. e. m RS Arc length of RS m RS e. = 360° 2 r 44 m RS = 2 (15.28) 360° 44 = m RS 360° 2 (15.28) 165° m RS EXAMPLE 4 EXAMPLE Find arc lengths to find measures Find the indicated measure. SOLUTION

16. 2. Length ofPQ 75° (9) 5.89 yd Arc length of PQ= 360° GUIDED PRACTICE Guided Practice Find the indicated measure.

17. 3. Circumference of N Arc length of LM =m LM 360° C 61.26 = 270° 360° C 61.26 = 27 36 C GUIDED PRACTICE Guided Practice 81.68 m = C

18. C=2 r 4. Radius of G = 2 3.12 r 25.2 Arc length of EF =m EF 2 3.14 r r = 360° C 25.2 10.5 = 150° 4.01 r 360° C 10.5 = 5 The radius of G is about 4.01. 12 C GUIDED PRACTICE Guided Practice 25.2 = C

19. The curves at the ends of the track shown are 180° arcs of circles. The radius of the arc for a runner on the red path shown is 36.8 meters. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter. EXAMPLE 5 EXAMPLE TRACK SOLUTION The path of a runner is made of two straight sections and two semicircles. To find the total distance, find the sum of the lengths of each part.

20. 1 = 2(84.39) + 2 2π 36.8 2 ANSWER The runner on the red path travels about 400 meters. EXAMPLE 5 Find arc lengths to find distance ≈ 400.0 meters

21. 1 = 2(84.39) + 2 2π 44.02 2 ANSWER The runner on the blue path travels about 445.4 meters. GUIDED PRACTICE Guided Practice 5. The radius of the arc for a runner on the blue path is 44.02 meters, as shown in the diagram. About how far does this runner travel to go once around the track? Round to the nearest tenth of a meter. ≈ 445.4 meters

22. Guided Practice Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. 6. FG Use formula for area of sector. Substitute 8 for r and 134 for m.  5.96 cm  18.71 cm Simplify.

23. Guided Practice Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. 7. 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.

24. =  m  4.19 m Guided Practice Find each arc length. Give your answer in terms of and rounded to the nearest hundredth. 8. GH Use formula for area of sector. Substitute 6 for r and 40 for m. Simplify.

25. Guided Practice Find each arc length. Give your answer in terms of and rounded to the nearest hundredth. 9. 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.

26. Homework Page 226-227 (#1 – 10, 17-21 all)