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Finding Formula for Circle Area via Triangle

Finding Formula for Circle Area via Triangle
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2017-09-09 06:41
TriangleCircleGeometric Construction
This example dynamically demonstrates how rotating a triangle forms a circle, visually illustrating the derivation of the circle area formula. Core math concepts include circle, triangle, rotation, Pythagorean theorem, etc. GeoGebra commands and objects like Min, Circle, Rotate, CircleArc, Segment are used in the construction. Suitable for high school mathematics teaching to help students understand the geometric meaning and derivation of the circle area formula.

GeoGebra Commands & Parameters

  1. Min(a - 1, -0.01) → a_1
  2. Circle(A, B) → c
  3. Rotate(B, ((2 * π) * x(B) / ((-x(A)) + x(B))), A) → B'
  4. CircleArc(A, B, B') → d
  5. Segment(B, C) → b
  6. CircleArc(A, B, B') → g
  7. Segment(B, C) → h
  8. Circle(A, B) → k
  9. Segment(A2, B2) → C2
  10. Rotate(A2, ((2 * π) * x(A2) / ((-x(A)) + x(A2))), A) → D2
  11. Rotate(A3, ((2 * π) * x(A3) / ((-x(A)) + x(A3))), A) → D3
  12. Rotate(A4, ((2 * π) * x(A4) / ((-x(A)) + x(A4))), A) → D4
  13. Rotate(A5, ((2 * π) * x(A5) / ((-x(A)) + x(A5))), A) → D5
  14. Rotate(A6, ((2 * π) * x(A6) / ((-x(A)) + x(A6))), A) → D6
  15. Rotate(A7, ((2 * π) * x(A7) / ((-x(A)) + x(A7))), A) → D7
  16. Rotate(A8, ((2 * π) * x(A8) / ((-x(A)) + x(A8))), A) → D8
  17. Rotate(A9, ((2 * π) * x(A9) / ((-x(A)) + x(A9))), A) → D9
  18. Rotate(A10, ((2 * π) * x(A10) / ((-x(A)) + x(A10))), A) → D10
  19. Rotate(A11, ((2 * π) * x(A11) / ((-x(A)) + x(A11))), A) → D11
  20. Rotate(A12, ((2 * π) * x(A12) / ((-x(A)) + x(A12))), A) → D12
  21. Rotate(A13, ((2 * π) * x(A13) / ((-x(A)) + x(A13))), A) → D13
  22. Rotate(A14, ((2 * π) * x(A14) / ((-x(A)) + x(A14))), A) → D14
  23. Rotate(A15, ((2 * π) * x(A15) / ((-x(A)) + x(A15))), A) → D15
  24. Rotate(A16, ((2 * π) * x(A16) / ((-x(A)) + x(A16))), A) → D16
  25. Rotate(A17, ((2 * π) * x(A17) / ((-x(A)) + x(A17))), A) → D17
  26. Rotate(A18, ((2 * π) * x(A18) / ((-x(A)) + x(A18))), A) → D18
  27. Rotate(A19, ((2 * π) * x(A19) / ((-x(A)) + x(A19))), A) → D19
  28. Rotate(A20, ((2 * π) * x(A20) / ((-x(A)) + x(A20))), A) → D20
  29. Rotate(A21, ((2 * π) * x(A21) / ((-x(A)) + x(A21))), A) → D21
  30. Rotate(A22, ((2 * π) * x(A22) / ((-x(A)) + x(A22))), A) → D22
  31. Rotate(A23, ((2 * π) * x(A23) / ((-x(A)) + x(A23))), A) → D23
  32. Rotate(A24, ((2 * π) * x(A24) / ((-x(A)) + x(A24))), A) → D24
  33. Rotate(A25, ((2 * π) * x(A25) / ((-x(A)) + x(A25))), A) → D25
  34. Rotate(A26, ((2 * π) * x(A26) / ((-x(A)) + x(A26))), A) → D26
  35. Rotate(A27, ((2 * π) * x(A27) / ((-x(A)) + x(A27))), A) → D27
  36. Rotate(A28, ((2 * π) * x(A28) / ((-x(A)) + x(A28))), A) → D28
  37. Rotate(A29, ((2 * π) * x(A29) / ((-x(A)) + x(A29))), A) → D29
  38. Rotate(A30, ((2 * π) * x(A30) / ((-x(A)) + x(A30))), A) → D30
  39. Rotate(A31, ((2 * π) * x(A31) / ((-x(A)) + x(A31))), A) → D31
  40. CircleArc(A, A2, D2) → E2
  41. CircleArc(A, A3, D3) → E3
  42. CircleArc(A, A4, D4) → E4
  43. CircleArc(A, A5, D5) → E5
  44. CircleArc(A, A6, D6) → E6
  45. CircleArc(A, A7, D7) → E7
  46. CircleArc(A, A8, D8) → E8
  47. CircleArc(A, A9, D9) → E9
  48. CircleArc(A, A10, D10) → E10
  49. CircleArc(A, A11, D11) → E11
  50. CircleArc(A, A12, D12) → E12
  51. CircleArc(A, A13, D13) → E13
  52. CircleArc(A, A14, D14) → E14
  53. CircleArc(A, A15, D15) → E15
  54. CircleArc(A, A16, D16) → E16
  55. CircleArc(A, A17, D17) → E17
  56. CircleArc(A, A18, D18) → E18
  57. CircleArc(A, A19, D19) → E19
  58. CircleArc(A, A20, D20) → E20
  59. CircleArc(A, A21, D21) → E21
  60. CircleArc(A, A22, D22) → E22
  61. CircleArc(A, A23, D23) → E23
  62. CircleArc(A, A24, D24) → E24
  63. CircleArc(A, A25, D25) → E25
  64. CircleArc(A, A26, D26) → E26
  65. CircleArc(A, A27, D27) → E27
  66. CircleArc(A, A28, D28) → E28
  67. CircleArc(A, A29, D29) → E29
  68. CircleArc(A, A30, D30) → E30
  69. CircleArc(A, A31, D31) → E31
  70. Segment(A3, B3) → C3
  71. Segment(A4, B4) → C4
  72. Segment(A5, B5) → C5
  73. Segment(A6, B6) → C6
  74. Segment(A7, B7) → C7
  75. Segment(A8, B8) → C8
  76. Segment(A9, B9) → C9
  77. Segment(A10, B10) → C10
  78. Segment(A11, B11) → C11
  79. Segment(A12, B12) → C12
  80. Segment(A13, B13) → C13
  81. Segment(A14, B14) → C14
  82. Segment(A15, B15) → C15
  83. Segment(A16, B16) → C16
  84. Segment(A17, B17) → C17
  85. Segment(A18, B18) → C18
  86. Segment(A19, B19) → C19
  87. Segment(A20, B20) → C20
  88. Segment(A21, B21) → C21
  89. Segment(A22, B22) → C22
  90. Segment(A23, B23) → C23
  91. Segment(A24, B24) → C24
  92. Segment(A25, B25) → C25
  93. Segment(A26, B26) → C26
  94. Segment(A27, B27) → C27
  95. Segment(A28, B28) → C28
  96. Segment(A29, B29) → C29
  97. Segment(A30, B30) → C30
  98. Segment(A31, B31) → C31
  99. Circle(A, A2) → F2
  100. Circle(A, A3) → F3
  101. Circle(A, A4) → F4
  102. Circle(A, A5) → F5
  103. Circle(A, A6) → F6
  104. Circle(A, A7) → F7
  105. Circle(A, A8) → F8
  106. Circle(A, A9) → F9
  107. Circle(A, A10) → F10
  108. Circle(A, A11) → F11
  109. Circle(A, A12) → F12
  110. Circle(A, A13) → F13
  111. Circle(A, A14) → F14
  112. Circle(A, A15) → F15
  113. Circle(A, A16) → F16
  114. Circle(A, A17) → F17
  115. Circle(A, A18) → F18
  116. Circle(A, A19) → F19
  117. Circle(A, A20) → F20
  118. Circle(A, A21) → F21
  119. Circle(A, A22) → F22
  120. Circle(A, A23) → F23
  121. Circle(A, A24) → F24
  122. Circle(A, A25) → F25
  123. Circle(A, A26) → F26
  124. Circle(A, A27) → F27
  125. Circle(A, A28) → F28
  126. Circle(A, A29) → F29
  127. Circle(A, A30) → F30
  128. Circle(A, A31) → F31