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# Develop the Area of a Circle Formula

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• ### Develop the Area of a Circle Formula

1. 1. How did theycome up with the formula for thearea of a circle?
2. 2. Gather the following• Print Circle.pdf
3. 3. Gather the following• Print Circle.pdf• If no access to a printer, get a circular object such as a cup or can, and piece of paper. Trace the circle onto the paper.
4. 4. Gather the following• Print Circle.pdf• If no access to a printer, get a circular object such as a cup or can, and piece of paper. Trace the circle onto the paper.• Scissors
5. 5. Gather the following• Print Circle.pdf• If no access to a printer, get a circular object such as a cup or can, and piece of paper. Trace the circle onto the paper.• Scissors• Ruler
6. 6. Do the following• Use your scissors to cut out the circle.
7. 7. Do the following• Use your scissors to cut out the circle.• Keep your scissors handy as you will need them.
8. 8. Do the following• Use your scissors to cut out the circle.• Keep your scissors handy as you will need them.• Work through the steps on the next several pages using your cut out circle.
9. 9. Do the following• Use your scissors to cut out the circle.• Keep your scissors handy as you will need them.• Work through the steps on the next several pages using your cut out circle.• When you see record, write the answer on your Learning Guide.
10. 10. (1) Fold the circle in half.
11. 11. (1) Fold the circle in half.(2) What does this fold represent?
12. 12. (1) Fold the circle in half.(2) What does this fold represent? ✓ The diameter.
13. 13. (1) Fold the circle in half.(2) What does this fold represent? ✓ The diameter.(3) Use the ruler to measure the fold. Record the measure of the fold.
14. 14. (1) Fold the circle in half.(2) What does this fold represent? ✓ The diameter.(3) Use the ruler to measure the fold. Record the measure of the fold.(4) Using the diameter, find the circumference of the circle and record the result.
15. 15. (1) Fold the circle in half.(2) What does this fold represent? ✓ The diameter.(3) Use the ruler to measure the fold. Record the measure of the fold.(4) Using the diameter, find the circumference of the circle and record the result.(5) Cut the circle on the fold to create 2 semi-circles.
16. 16. (6) Fold each semi-circle in half.
17. 17. (6) Fold each semi-circle in half.(7) What does this fold represent?
18. 18. (6) Fold each semi-circle in half.(7) What does this fold represent? ✓ The radius.
19. 19. (6) Fold each semi-circle in half.(7) What does this fold represent? ✓ The radius.(8) Record the measure of the fold.
20. 20. (6) Fold each semi-circle in half.(7) What does this fold represent? ✓ The radius.(8) Record the measure of the fold.(9) Cut the semi-circles on the folds. You should have 4 pieces now.
21. 21. (6) Fold each semi-circle in half.(7) What does this fold represent? ✓ The radius.(8) Record the measure of the fold.(9) Cut the semi-circles on the folds. You should have 4 pieces now.(10) Each piece represents what fraction of the circle?
22. 22. (6) Fold each semi-circle in half.(7) What does this fold represent? ✓ The radius.(8) Record the measure of the fold.(9) Cut the semi-circles on the folds. You should have 4 pieces now.(10) Each piece represents what fraction of the circle? ✓ 1/4
23. 23. (11)Fold each of the 4 pieces in half.
24. 24. (11)Fold each of the 4 pieces in half.(12)Record the measure of the fold.
25. 25. (11)Fold each of the 4 pieces in half.(12)Record the measure of the fold.(13)Record your observation between this fold to the other 2 folds.
26. 26. (11)Fold each of the 4 pieces in half.(12)Record the measure of the fold.(13)Record your observation between this fold to the other 2 folds.(14)What does this fold represent?
27. 27. (11)Fold each of the 4 pieces in half.(12)Record the measure of the fold.(13)Record your observation between this fold to the other 2 folds.(14)What does this fold represent? ✓ The radius.
28. 28. (11)Fold each of the 4 pieces in half.(12)Record the measure of the fold.(13)Record your observation between this fold to the other 2 folds.(14)What does this fold represent? ✓ The radius.(15)Cut the 4 pieces on the folds. You should now have 8 pieces.
29. 29. (16)Each piece represents what fraction of the circle?
30. 30. (16)Each piece represents what fraction of the circle? ✓ 1/8
31. 31. (16)Each piece represents what fraction of the circle? ✓ 1/8(17)Fold each of the 8 pieces in half.
32. 32. (16)Each piece represents what fraction of the circle? ✓ 1/8(17)Fold each of the 8 pieces in half.(18)Record the measure of the fold.
33. 33. (16)Each piece represents what fraction of the circle? ✓ 1/8(17)Fold each of the 8 pieces in half.(18)Record the measure of the fold.(19)Record your observation between this fold to the other 3 folds.
34. 34. (20)What does this fold represent?
35. 35. (20)What does this fold represent? ✓ The radius.
36. 36. (20)What does this fold represent? ✓ The radius.(21)Cut the 8 pieces on the folds. You should have 16 pieces.
37. 37. (20)What does this fold represent? ✓ The radius.(21)Cut the 8 pieces on the folds. You should have 16 pieces.(22)Each piece represents what fraction of the circle?
38. 38. (20)What does this fold represent? ✓ The radius.(21)Cut the 8 pieces on the folds. You should have 16 pieces.(22)Each piece represents what fraction of the circle? ✓ 1/16
39. 39. (23)Arrange the pieces of the circle to make a rectangle best you can.
40. 40. (23)Arrange the pieces of the circle to make a rectangle best you can.(24)Record the measure of the base and height of the rectangle created.
41. 41. (25) Record your observations between the measure of the base and height of the rectangle and previous measures recorded or calculated.
42. 42. (25) Record your observations between the measure of the base and height of the rectangle and previous measures recorded or calculated.(26)How do you find the area of a rectangle?
43. 43. (25) Record your observations between the measure of the base and height of the rectangle and previous measures recorded or calculated.(26)How do you find the area of a rectangle? ✓ Area = base * height
44. 44. (25) Record your observations between the measure of the base and height of the rectangle and previous measures recorded or calculated.(26)How do you find the area of a rectangle? ✓ Area = base * height(27)Notice the height of the rectangle is one of the cuts made. What is the relationship between the cut and the circle?
46. 46. ✓ The radius.(28)What part of the circle makes up the longer sides of the rectangle?
47. 47. ✓ The radius.(28)What part of the circle makes up the longer sides of the rectangle? ✓ The circumference.
48. 48. ✓ The radius.(28)What part of the circle makes up the longer sides of the rectangle? ✓ The circumference.(29)How do you find the circumference of a circle?
49. 49. ✓ The radius.(28)What part of the circle makes up the longer sides of the rectangle? ✓ The circumference.(29)How do you find the circumference of a circle? ✓ C = 2πr or
50. 50. ✓ The radius.(28)What part of the circle makes up the longer sides of the rectangle? ✓ The circumference.(29)How do you find the circumference of a circle? ✓ C = 2πr or ✓ C = πd
51. 51. (30)To find the area of a rectangle, only one of the 2 bases (or longer sides) is needed.
52. 52. (30)To find the area of a rectangle, only one of the 2 bases (or longer sides) is needed.(31)If C = 2πr is used to find the circumference of the whole circle, how can you find the length of one base?
53. 53. (30)To find the area of a rectangle, only one of the 2 bases (or longer sides) is needed.(31)If C = 2πr is used to find the circumference of the whole circle, how can you find the length of one base? ✓ Divide the circumference by 2 to represent half the diameter.
54. 54. Recap what we have✓ The height of the rectangle is the radius of the original circle.
55. 55. Recap what we have✓ The height of the rectangle is the radius of the original circle.✓ The base of the rectangle is half the circumference of the original circle.
56. 56. Recap what we have✓ The height of the rectangle is the radius of the original circle.✓ The base of the rectangle is half the circumference of the original circle.✓ The area of a rectangle is found by multiplying the base times the height of the rectangle.
57. 57. Recap what we have✓ The height of the rectangle is the radius of the original circle.✓ The base of the rectangle is half the circumference of the original circle.✓ The area of a rectangle is found by multiplying the base times the height of the rectangle.✓ Substitute what was found for the base and height to find the formula for the area of a circle.
58. 58. Area of a Circle Formula A = b⋅h • Start with the formula for the area of a rectangle.
59. 59. Area of a Circle Formula A = b⋅h • Start with the formula for the area of a rectangle. • Substitute the base as half the circumference.
60. 60. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 • Substitute the base as half the circumference.
61. 61. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 • Substitute the base as half the circumference. • Substitute the radius for the height.
62. 62. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 1 • Substitute the base as A = ⋅ 2π r ⋅ r 2 half the circumference. • Substitute the radius for the height.
63. 63. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 1 • Substitute the base as A = ⋅ 2π r ⋅ r 2 half the circumference. • Substitute the radius for the height. • Simplify.
64. 64. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 1 • Substitute the base as A = ⋅ 2π r ⋅ r 2 half the circumference. 1 1+1 • Substitute the radius A = ⋅ 2π r 2 for the height. • Simplify.
65. 65. Area of a Circle Formula A = b⋅h • Start with the formula 1 for the area of a A = ⋅ 2π r ⋅ h rectangle. 2 1 • Substitute the base as A = ⋅ 2π r ⋅ r 2 half the circumference. 1 1+1 • Substitute the radius A = ⋅ 2π r 2 for the height. 2 A = πr • Simplify.
66. 66. Area of a Circle = πr 2