8 radian arc length and area formulas

8 Radian Arc Length and Area Formulas

1. Radian–Measurement Applications  r = 1 The unit circle is the circle centered at (0, 0) with radius 1. (1, 0)

2. Radian–Measurement Applications  r = 1 The unit circle is the circle centered at (0, 0) with radius 1. (1, 0) It's the graph of x 2 + y 2 = 1 .

3. Radian–Measurement Applications  r = 1 The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  (1, 0)

4. Radian–Measurement Applications  r = 1 The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. (1, 0)

5. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. (1, 0)

6. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 Conversions between Degree and Radian The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. 180 o = π rad (1, 0)

7. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 Conversions between Degree and Radian The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. 180 o = π rad 90 o = rad π 2 (1, 0)

8. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 Conversions between Degree and Radian The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. 180 o = π rad 90 o = rad π 2 60 o = rad π 3 45 o = rad π 4 (1, 0)

9. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 Conversions between Degree and Radian π 180 π 180 o The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. 1 o =  0.0175 rad 1 rad =  57 o ‘ 180 o = π rad 90 o = rad π 2 60 o = rad π 3 45 o = rad π 4 (1, 0)

10. Arc length as angle measurement for  Radian–Measurement Applications  r = 1 Conversions between Degree and Radian π 180 π 180 o The unit circle is the circle centered at (0, 0) with radius 1. It's the graph of x 2 + y 2 = 1 . The radian measurement of the angle  is the length of the arc that the angle  cuts off from the unit circle. 1 o =  0.0175 rad 1 rad =  57 o ‘ 180 o = π rad 90 o = rad π 2 60 o = rad π 3 45 o = rad π 4 (1, 0) The advantage of using circular–lengths (radians) to measure angles is that formulas concerning circles may be stated with greater simplicity.

11. An angle  based at the center of a circle is called a central angle. r Arc Length Formula 

12. Radian Arc-Length Formula An angle  based at the center of a circle is called a central angle. r Arc Length Formula 

13. Radian Arc-Length Formula Given a circle of radius = r and a central angle  in radian , let L = length of the arc cuts off by the angle  , An angle  based at the center of a circle is called a central angle. r Arc Length Formula 

14. Radian Arc-Length Formula Given a circle of radius = r and a central angle  in radian , let L = length of the arc cuts off by the angle  , then L = r  An angle  based at the center of a circle is called a central angle. r Arc Length Formula  L = r  L= arc length

15. Radian Arc-Length Formula Given a circle of radius = r and a central angle  in radian , let L = length of the arc cuts off by the angle  , then L = r  An angle  based at the center of a circle is called a central angle. r Arc Length Formula  L = r  L= arc length This formula is based on the following proportion which demonstrates the advantage of using radian.

16. Radian Arc-Length Formula Given a circle of radius = r and a central angle  in radian , let L = length of the arc cuts off by the angle  , then L = r  An angle  based at the center of a circle is called a central angle. r Arc Length Formula  L = r  L= arc length This formula is based on the following proportion which demonstrates the advantage of using radian. arc length : circumference = L : 2 π r =  : 2 π (in rad) 2 π r L = 2 π  or

17. Radian Arc-Length Formula Given a circle of radius = r and a central angle  in radian , let L = length of the arc cuts off by the angle  , then L = r  An angle  based at the center of a circle is called a central angle. r Arc Length Formula  L = r  L= arc length This formula is based on the following proportion which demonstrates the advantage of using radian. arc length : circumference = L : 2 π r =  : 2 π (in rad) 2 π r L = 2 π  or clear denominators, we´ve L = r 

18. Arc Length Formula Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust?

19. Arc Length Formula Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? 50 o 18 ? a.

20. Arc Length Formula Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian! 50 o 18 ? a.

21. Arc Length Formula Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, 50 o 18 ? a.

22. Arc Length Formula Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? a.

23. Arc Length Formula b. A slice of pizza cut from a pizza with 32” diameter has a crust measured 12”. Find its central angle. Give the answer in radian and degree. Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? 16 ? 12 a. b.

24. Arc Length Formula Given that r = 16" and L = 12", therefore 16·  = 12 b. A slice of pizza cut from a pizza with 32” diameter has a crust measured 12”. Find its central angle. Give the answer in radian and degree. Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? 16 ? 12 a. b.

25. Arc Length Formula Given that r = 16" and L = 12", therefore 16·  = 12 or that  = 12/16 = ¾ rad b. A slice of pizza cut from a pizza with 32” diameter has a crust measured 12”. Find its central angle. Give the answer in radian and degree. Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? 16 ? 12 a. b.

26. Arc Length Formula Given that r = 16" and L = 12", therefore 16·  = 12 or that  = 12/16 = ¾ rad b. A slice of pizza cut from a pizza with 32” diameter has a crust measured 12”. Find its central angle. Give the answer in radian and degree. 180 o π i n degree,  = ¾ rad = ¾ * Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? 16 ? 12 a. b.

27. Arc Length Formula Given that r = 16" and L = 12", therefore 16·  = 12 or that  = 12/16 = ¾ rad b. A slice of pizza cut from a pizza with 32” diameter has a crust measured 12”. Find its central angle. Give the answer in radian and degree. 180 o π 135 o π i n degree,  = ¾ rad = ¾ * = ≈ 43.0 o Example A. a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the length of its crust? The point here is that we need radian ! 50 o = 50 * ( π /180 rad) = 5 π /18 rad, with r = 18, the length of the crust is r  = 18· 5 π 18 = 5 π  15.7" 50 o 18 ? 16 ? 12 a. b.

28. r  Area Formula Radian Area Formula

29. r  Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2

30. r  A = area Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 A= r 2  1 2

31. r  A = area Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 A= r 2  1 2 Example B . a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the area of the slice?

32. r  A = area Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 A= r 2  1 2 Example B . a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the area of the slice?

33. r  A = area Converting degree to radian 50 o = 5 π /18 rad, Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 A= r 2  1 2 Example B . a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the area of the slice?

34. r  A = area Converting degree to radian 50 o = 5 π /18 rad, with r = 18, the area of the slice is 5 π 18 Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 1 2 A= r 2  1 2 r 2  = * 18 2 * 1 2 Example B . a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the area of the slice?

35. r  A = area Converting degree to radian 50 o = 5 π /18 rad, with r = 18, the area of the slice is 5 π 18 Area Formula Radian Area Formula Given a circle of radius = r, and a central angle =  in radian , the area A of the slice cut out by  is r 2  /2 , i.e. A = r 2  1 2 1 2 A= r 2  1 2 r 2  = * 18 2 * 1 2 Example B . a. A slice of pizza with central angle of 50 o is cut from a 36”-diamter pizza, what is the area of the slice? = 45 π  141 in 2

36. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust?

37. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust?

38. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  .

39. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2

40. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad = 

41. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad =  Therefore L = r  = 8* ¾ = 6"

42. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad =  Therefore L = r  = 8* ¾ = 6" Example D. A slice of pizza with 12-inch crust is cut from a 16-inch diameter pizza, what is its area?

43. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad =  Therefore L = r  = 8* ¾ = 6" Example D. A slice of pizza with 12-inch crust is cut from a 16-inch diameter pizza, what is its area? The arc length L = 12, r = 8, and L = r  , hence12 = 8 

44. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad =  Therefore L = r  = 8* ¾ = 6" Example D. A slice of pizza with 12-inch crust is cut from a 16-inch diameter pizza, what is its area? The arc length L = 12, r = 8, and L = r  , hence12 = 8  or 3/2 rad = 

45. Area and Arc Length Example C. A 24-inch 2 slice of pizza is cut from a 16-inch diameter pizza, what is the length of its crust? Need the angle  . A = 24, r = 8, and A = r 2  /2 hence 24 = 8 2  /2 = 64  /2 24 = 32   ¾ rad =  Therefore L = r  = 8* ¾ = 6" Example D. A slice of pizza with 12-inch crust is cut from a 16-inch diameter pizza, what is its area? The arc length L = 12, r = 8, and L = r  , hence12 = 8  or 3/2 rad =  So A = r 2  /2 = 8 2 * * = 48 in 2 1 2 3 2

46. The simplicity of these formulas is carried over to other formulas concerning rotations. Angular Velocity

47. The simplicity of these formulas is carried over to other formulas concerning rotations. For example, t he angular velocity w of a rotating wheel is the amount of angle rotated in one unit of time. Angular Velocity

48. The simplicity of these formulas is carried over to other formulas concerning rotations. For example, t he angular velocity w of a rotating wheel is the amount of angle rotated in one unit of time. The angular velocity w = ( π /2) rad /sec means the wheel rotates ¼ of a round (circle) every second. Angular Velocity

49. The simplicity of these formulas is carried over to other formulas concerning rotations. For example, t he angular velocity w of a rotating wheel is the amount of angle rotated in one unit of time. The angular velocity w = ( π /2) rad /sec means the wheel rotates ¼ of a round (circle) every second. The angular velocity w = ( π /2) / sec Assuming t is in second and w is in radian, then the blue dot travels the arc length of w*r every second. Angular Velocity r

50. The simplicity of these formulas is carried over to other formulas concerning rotations. For example, t he angular velocity w of a rotating wheel is the amount of angle rotated in one unit of time. The angular velocity w = ( π /2) rad /sec means the wheel rotates ¼ of a round (circle) every second. The angular velocity w = ( π /2) / sec Assuming t is in second and w is in radian, then the blue dot travels the arc length of w*r every second. So in t seconds, the linear distance D or the distance the wheel traveled on the ground is D = w*r*t Angular Velocity D=w*r*t r

51. The simplicity of these formulas is carried over to other formulas concerning rotations. For example, t he angular velocity w of a rotating wheel is the amount of angle rotated in one unit of time. The angular velocity w = ( π /2) rad /sec means the wheel rotates ¼ of a round (circle) every second. The angular velocity w = ( π /2) / sec Assuming t is in second and w is in radian, then the blue dot travels the arc length of w*r every second. So in t seconds, the linear distance D or the distance the wheel traveled on the ground is D = w*r*t and the dial have swiped over an area of A = ½ w*r 2 *t . Angular Velocity D=w*r*t A =½ w*r 2 *t r

52. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? Angular Velocity 5 w = 4 π rad /sec

53. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? Angular Velocity 5 w = 4 π rad /sec Its linear speed is 4 π (5) = 20 π m /sec

54. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? Angular Velocity 5 w = 4 π rad /sec Its linear speed is 4 π (5) = 20 π m /sec There are 60 seconds in one minutes so the distance it traveled is D = w*r*t = 4 π (5)(60) = 1200 π m .

55. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? b. How much distance does the point p on the sphere as shown travel in one minute? What is its linear speed ? 60 o p Angular Velocity Its linear speed is 4 π (5) = 20 π m /sec There are 60 seconds in one minutes so the distance it traveled is D = w*r*t = 4 π (5)(60) = 1200 π m . 5 w = 4 π rad /sec

56. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? b. How much distance does the point p on the sphere as shown travel in one minute? What is its linear speed ? 60 o p Angular Velocity Its linear speed is 4 π (5) = 20 π m /sec There are 60 seconds in one minutes so the distance it traveled is D = w*r*t = 4 π (5)(60) = 1200 π m . 5 r The radius of the rotation is r = 5 sin ( 30 o ) = 5/2 meters. w = 4 π rad /sec

57. Example D. A sphere with radius r = 5 meters is spinning with the angular velocity w = 4 π rad /sec. a. What is its linear speed? How much distance does a point travel along the equator in one minute? b. How much distance does the point p on the sphere as shown travel in one minute? What is its linear speed ? 60 o p Angular Velocity Its linear speed is 4 π (5) = 20 π m /sec There are 60 seconds in one minutes so the distance it traveled is D = w*r*t = 4 π (5)(60) = 1200 π m . 5 r The radius of the rotation is r = 5 sin ( 30 o ) = 5/2 meters. So p travels 600 π at a linear speed of 1 π m /sec. w = 4 π rad /sec

58. Exercise Radian–Measurement Applications At Pizza Grande, a medium pizza has 12–inch diameter and a large pizza has 18–inch diameter. A large pizza is cut into 8 slices sold at $3/slice and a medium one is cut into 6 slices and sold at $2/slice. 1. Find the perimeter and the area of a medium slice . 2. Find the perimeter and the area of a large slice . 3. Which is a better deal, a medium or a large slice? 4. We want to cut one slice from the medium pizza that is the size of two large slices. What is the central angle of the medium–slice? 5. We want to cut one slice from the large pizza that is the size as three medium slices . What is the central angle of the large–slice?

59. 6. A 25 in 2 slice of pizza has a 8" crust. How much is the rest of the pizza 7. A 25 in 2 slice of pizza has a 35 o central angle. How much is the rest of the pizza 8. A slice of pizza has a 8" crust and a 35 o central angle. How much is the rest of the pizza 9. A slice of pizza cut from a pizza with 9" radius has a 8 " crust. What is the central angle of the slice? How much is the rest of the pizza Radian–Measurement Applications 10 . A car has 18”–radius wheels is traveling with the angular velocity of w = 10 π rad /sec . How fast is car traveling in mph?

60. Radian–Measurement Applications 11. A car has 15”–radius wheels, w hat is the approx. angular velocity (rad/sec) of the wheels when it’s traveling at a speed of 60 mph ? 12. A radar spins at rate of w = π /4 rad /sec and has a 20–mile radius effective detection area. In 3 second, how much area is scanned by the radar? 13. From problem 12, how long would it take for the radar to scanned an area of 100 mi 2 ? ≈ 8000 mi Tropic of Cancer Arctic Circle ≈ 23 o ≈ 66 o 14. Following are approximate measurements of earth. Find the linear speeds in mph at the equator, at the Tropic of Cancer and at the Arctic C ircle.

61. A nswers Radian–Measurement Applications 1. p = 2 π , A = 6 π 3. a large slice is a better deal 5 . 4 π /9 rad 7. 232 in 2 9. 218 in 2 11. w ≈ 70.4 rad/sec 13. t = 2/ π sec

62. Radian–Measurement Applications 11. A car has 15”–radius wheels, w hat is the approx. angular velocity (rad/sec) of the wheels when it’s traveling at a speed of 60 mph ? 12. A radar spins at rate of w = π /4 rad /sec and has a 200–mile radius effective detection area. In 3 second, how much area is scanned by the radar? 13. From problem 11, how long would it take for the radar to scanned an area of 100 mi 2 ? ≈ 8000 mi Tropic of Cancer Arctic Circle ≈ 23 o ≈ 66 o 14. Following are approximate measurements of earth. Find the linear speeds in mph at the equator, at the Tropic of Cancer and at the Arctic C ircle.