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Area of Circle as Parallelogram Transformation

Area of Circle as Parallelogram Transformation
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2020-12-19 16:04
CircleVectorGeometric Construction
This example demonstrates how to transform the area of a circle into the area of a parallelogram. Key mathematical concepts include circles, parallelograms, and geometric transformations. GeoGebra commands used include Min, Intersect, Point, Rotate, Vector, and Translate. Suitable for middle school and above students learning geometric transformations and area calculations.

GeoGebra Commands & Parameters

  1. Min(i, 1) → C1
  2. Intersect(xAxis, yAxis) → A
  3. Point(yAxis) → B
  4. Rotate(B, (360 * ° / n), A) → B'_L
  5. Rotate(B, (-360 * ° / n), A) → B'_R
  6. Rotate(B3, B5 / 2, B7) → C7
  7. Rotate(B7, (-B$4), C7) → B8
  8. Rotate(C7, B$5, B8) → C8
  9. Rotate(B8, (-B$4), C8) → B9
  10. Rotate(C8, B$5, B9) → C9
  11. Rotate(B9, (-B$4), C9) → B10
  12. Rotate(C9, B$5, B10) → C10
  13. Rotate(B10, (-B$4), C10) → B11
  14. Rotate(C10, B$5, B11) → C11
  15. Rotate(B11, (-B$4), C11) → B12
  16. Rotate(C11, B$5, B12) → C12
  17. Rotate(B12, (-B$4), C12) → B13
  18. Rotate(C12, B$5, B13) → C13
  19. Rotate(B13, (-B$4), C13) → B14
  20. Rotate(C13, B$5, B14) → C14
  21. Rotate(B14, (-B$4), C14) → B15
  22. Rotate(C14, B$5, B15) → C15
  23. Rotate(B15, (-B$4), C15) → B16
  24. Rotate(C15, B$5, B16) → C16
  25. Rotate(B16, (-B$4), C16) → B17
  26. Rotate(C16, B$5, B17) → C17
  27. Rotate(B17, (-B$4), C17) → B18
  28. Rotate(C17, B$5, B18) → C18
  29. Rotate(B18, (-B$4), C18) → B19
  30. Rotate(C18, B$5, B19) → C19
  31. Rotate(B19, (-B$4), C19) → B20
  32. Rotate(C19, B$5, B20) → C20
  33. Rotate(B20, (-B$4), C20) → B21
  34. Rotate(C20, B$5, B21) → C21
  35. Rotate(B21, (-B$4), C21) → B22
  36. Rotate(C21, B$5, B22) → C22
  37. Rotate(B22, (-B$4), C22) → B23
  38. Rotate(C22, B$5, B23) → C23
  39. Rotate(B23, (-B$4), C23) → B24
  40. Rotate(C23, B$5, B24) → C24
  41. Rotate(B24, (-B$4), C24) → B25
  42. Rotate(C24, B$5, B25) → C25
  43. Rotate(B25, (-B$4), C25) → B26
  44. Rotate(C25, B$5, B26) → C26
  45. Rotate(B26, (-B$4), C26) → B27
  46. Rotate(C26, B$5, B27) → C27
  47. Vector(B7, C7) → u
  48. Translate(CircleSector[C7, B8, B7], Vector[(C$2 * u)]) → D7
  49. Translate(CircleSector[C8, B9, B8], Vector[(C$2 * u)]) → D8
  50. Translate(CircleSector[F7, E7, E8], Vector[(C$2 * u)]) → G7
  51. Translate(CircleSector[F8, E8, E9], Vector[(C$2 * u)]) → G8
  52. Translate(CircleSector[I7, H7, H8], Vector[((-C$2) * u)]) → J7
  53. Translate(CircleSector[L7, K8, K7], Vector[((-C$2) * u)]) → M7
  54. Translate(CircleSector[L8, K9, K8], Vector[((-C$2) * u)]) → M8
  55. Translate(CircleSector[I8, H8, H9], Vector[((-C$2) * u)]) → J8
  56. Translate(CircleSector[C9, B10, B9], Vector[(C$2 * u)]) → D9
  57. Translate(CircleSector[F9, E9, E10], Vector[(C$2 * u)]) → G9
  58. Translate(CircleSector[L9, K10, K9], Vector[((-C$2) * u)]) → M9
  59. Translate(CircleSector[I9, H9, H10], Vector[((-C$2) * u)]) → J9
  60. Translate(CircleSector[C10, B11, B10], Vector[(C$2 * u)]) → D10
  61. Translate(CircleSector[F10, E10, E11], Vector[(C$2 * u)]) → G10
  62. Translate(CircleSector[L10, K11, K10], Vector[((-C$2) * u)]) → M10
  63. Translate(CircleSector[I10, H10, H11], Vector[((-C$2) * u)]) → J10
  64. Translate(CircleSector[C11, B12, B11], Vector[(C$2 * u)]) → D11
  65. Translate(CircleSector[F11, E11, E12], Vector[(C$2 * u)]) → G11
  66. Translate(CircleSector[L11, K12, K11], Vector[((-C$2) * u)]) → M11
  67. Translate(CircleSector[I11, H11, H12], Vector[((-C$2) * u)]) → J11
  68. Translate(CircleSector[C12, B13, B12], Vector[(C$2 * u)]) → D12
  69. Translate(CircleSector[F12, E12, E13], Vector[(C$2 * u)]) → G12
  70. Translate(CircleSector[L12, K13, K12], Vector[((-C$2) * u)]) → M12
  71. Translate(CircleSector[I12, H12, H13], Vector[((-C$2) * u)]) → J12
  72. Translate(CircleSector[C13, B14, B13], Vector[(C$2 * u)]) → D13
  73. Translate(CircleSector[F13, E13, E14], Vector[(C$2 * u)]) → G13
  74. Translate(CircleSector[L13, K14, K13], Vector[((-C$2) * u)]) → M13
  75. Translate(CircleSector[I13, H13, H14], Vector[((-C$2) * u)]) → J13
  76. Translate(CircleSector[C14, B15, B14], Vector[(C$2 * u)]) → D14
  77. Translate(CircleSector[F14, E14, E15], Vector[(C$2 * u)]) → G14
  78. Translate(CircleSector[L14, K15, K14], Vector[((-C$2) * u)]) → M14
  79. Translate(CircleSector[I14, H14, H15], Vector[((-C$2) * u)]) → J14
  80. Translate(CircleSector[C15, B16, B15], Vector[(C$2 * u)]) → D15
  81. Translate(CircleSector[F15, E15, E16], Vector[(C$2 * u)]) → G15
  82. Translate(CircleSector[L15, K16, K15], Vector[((-C$2) * u)]) → M15
  83. Translate(CircleSector[I15, H15, H16], Vector[((-C$2) * u)]) → J15
  84. Translate(CircleSector[C16, B17, B16], Vector[(C$2 * u)]) → D16
  85. Translate(CircleSector[F16, E16, E17], Vector[(C$2 * u)]) → G16
  86. Translate(CircleSector[L16, K17, K16], Vector[((-C$2) * u)]) → M16
  87. Translate(CircleSector[I16, H16, H17], Vector[((-C$2) * u)]) → J16
  88. Translate(CircleSector[C17, B18, B17], Vector[(C$2 * u)]) → D17
  89. Translate(CircleSector[F17, E17, E18], Vector[(C$2 * u)]) → G17
  90. Translate(CircleSector[L17, K18, K17], Vector[((-C$2) * u)]) → M17
  91. Translate(CircleSector[I17, H17, H18], Vector[((-C$2) * u)]) → J17
  92. Translate(CircleSector[C18, B19, B18], Vector[(C$2 * u)]) → D18
  93. Translate(CircleSector[F18, E18, E19], Vector[(C$2 * u)]) → G18
  94. Translate(CircleSector[L18, K19, K18], Vector[((-C$2) * u)]) → M18
  95. Translate(CircleSector[I18, H18, H19], Vector[((-C$2) * u)]) → J18
  96. Translate(CircleSector[C19, B20, B19], Vector[(C$2 * u)]) → D19
  97. Translate(CircleSector[F19, E19, E20], Vector[(C$2 * u)]) → G19
  98. Translate(CircleSector[L19, K20, K19], Vector[((-C$2) * u)]) → M19
  99. Translate(CircleSector[I19, H19, H20], Vector[((-C$2) * u)]) → J19
  100. Translate(CircleSector[C20, B21, B20], Vector[(C$2 * u)]) → D20
  101. Translate(CircleSector[F20, E20, E21], Vector[(C$2 * u)]) → G20
  102. Translate(CircleSector[L20, K21, K20], Vector[((-C$2) * u)]) → M20
  103. Translate(CircleSector[I20, H20, H21], Vector[((-C$2) * u)]) → J20
  104. Translate(CircleSector[C21, B22, B21], Vector[(C$2 * u)]) → D21
  105. Translate(CircleSector[F21, E21, E22], Vector[(C$2 * u)]) → G21
  106. Translate(CircleSector[L21, K22, K21], Vector[((-C$2) * u)]) → M21
  107. Translate(CircleSector[I21, H21, H22], Vector[((-C$2) * u)]) → J21
  108. Translate(CircleSector[C22, B23, B22], Vector[(C$2 * u)]) → D22
  109. Translate(CircleSector[F22, E22, E23], Vector[(C$2 * u)]) → G22
  110. Translate(CircleSector[L22, K23, K22], Vector[((-C$2) * u)]) → M22
  111. Translate(CircleSector[I22, H22, H23], Vector[((-C$2) * u)]) → J22
  112. Translate(CircleSector[C23, B24, B23], Vector[(C$2 * u)]) → D23
  113. Translate(CircleSector[F23, E23, E24], Vector[(C$2 * u)]) → G23
  114. Translate(CircleSector[L23, K24, K23], Vector[((-C$2) * u)]) → M23
  115. Translate(CircleSector[I23, H23, H24], Vector[((-C$2) * u)]) → J23
  116. Translate(CircleSector[C24, B25, B24], Vector[(C$2 * u)]) → D24
  117. Translate(CircleSector[F24, E24, E25], Vector[(C$2 * u)]) → G24
  118. Translate(CircleSector[L24, K25, K24], Vector[((-C$2) * u)]) → M24
  119. Translate(CircleSector[I24, H24, H25], Vector[((-C$2) * u)]) → J24
  120. Translate(CircleSector[C25, B26, B25], Vector[(C$2 * u)]) → D25
  121. Translate(CircleSector[F25, E25, E26], Vector[(C$2 * u)]) → G25
  122. Translate(CircleSector[L25, K26, K25], Vector[((-C$2) * u)]) → M25
  123. Translate(CircleSector[I25, H25, H26], Vector[((-C$2) * u)]) → J25
  124. Translate(CircleSector[C26, B27, B26], Vector[(C$2 * u)]) → D26
  125. Translate(CircleSector[F26, E26, E27], Vector[(C$2 * u)]) → G26
  126. Translate(CircleSector[L26, K27, K26], Vector[((-C$2) * u)]) → M26
  127. Translate(CircleSector[I26, H26, H27], Vector[((-C$2) * u)]) → J26

AI Drawing Prompt Reference

Draw a circle and transform its area into a parallelogram. Start with coordinate axes and a point on the y-axis as the radius endpoint. Rotate this point around the center by equal angular steps (360 degrees divided by the number of sectors) to generate a series of sector vertices, connecting adjacent vertices to form triangular sectors. Then translate the first sector to align with the others, assembling them into a parallelogram-like shape. Include the circle, the rotated sectors, the translation vector, and the final parallelogram outline to illustrate the area conversion.
The prompt above was compiled from the original drawing commands and is for reference only.