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Circle Area Formula Animation (Parallelogram)

Circle Area Formula Animation (Parallelogram)
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2016-06-27 20:15
CirclePolygonGeometric Construction
This example demonstrates how to derive the area of a circle by approximating it with a parallelogram through animation. Key math concepts include circle, area, parallelogram, multiplication, linear equations, and step functions. Construction highlights the use of GeoGebra commands and objects like Circle, Point, Vector, Translate, Rotate, Line, Angle, Mirror, Polygon, OrthogonalLine, Segment. Suitable for middle school and above to understand the derivation of the circle area formula.

GeoGebra Commands & Parameters

  1. Circle(A, a) → c
  2. Point(c) → B
  3. Vector(B, A) → u
  4. Translate(C, Vector[(-u)]) → C'_2
  5. Rotate(C'_2, ((g / 100 * 15))°, C'_2) → C''_2
  6. Line(B, A) → b
  7. Rotate(B, 30*°, A) → B'
  8. Angle(B, A, B') → α
  9. Line(B', A) → d
  10. Circle(C, a) → e
  11. Vector(A, B') → v
  12. Vector(B, B') → w
  13. Translate(C, v) → C'_1
  14. Rotate(C'_2, 60*°, C) → C''_1
  15. Rotate(C'_1, 60*°, C) → D
  16. Line(C'_2, C) → f
  17. Mirror(C'_1, f) → E
  18. Mirror(C''_1, f) → C'''
  19. Mirror(D, f) → D'
  20. Polygon(E, C'_2, C) → poly1, c_1, e_1, c'
  21. Rotate(E, ((g / 100 * 15))°, C'_2) → E'_1
  22. Rotate(C, ((g / 100 * 15))°, C'_2) → C'_3
  23. Polygon(E'_1, C''_2, C'_3) → poly1', c'_1, e', c''
  24. Polygon(C''', E, C) → poly2, c_2, c''', e_2
  25. Rotate(C''', (g / 100 * β), F) → G
  26. Rotate(E, (g / 100 * β), F) → E'
  27. Rotate(C, (g / 100 * β), F) → C'_4
  28. Polygon(G, E', C'_4) → poly2', c'_2, g_1, e'_1
  29. Polygon(C, D', C''') → poly3, c'''_1, c_3, d'
  30. Rotate(C, (g / 100 * γ), H) → C'
  31. Rotate(D', (g / 100 * γ), H) → D''
  32. Rotate(C''', (g / 100 * γ), H) → I
  33. Polygon(C', D'', I) → poly3', i, c'_3, d''
  34. Mirror(C', f) → C''_3
  35. Mirror(D'', f) → D'''_1
  36. Mirror(I, f) → I'_1
  37. Polygon(C''_3, D'''_1, I'_1) → poly3''_1, i', c''_1, d'''
  38. Mirror(G, f) → G'_1
  39. Mirror(E', f) → E''_1
  40. Mirror(C'_4, f) → J
  41. Polygon(G'_1, E''_1, J) → poly2''_1, j, g', e''
  42. Mirror(E'_1, f) → K
  43. Mirror(C''_2, f) → L
  44. Mirror(C'_3, f) → M
  45. Polygon(K, L, M) → poly1''_1, m, k, l
  46. OrthogonalLine(C, f) → h
  47. Mirror(C', h) → C''
  48. Mirror(D'', h) → D'''
  49. Mirror(I, h) → I'
  50. Polygon(C'', D''', I') → poly3'', i'_1, c''_2, d'''_1
  51. Mirror(G, h) → G'
  52. Mirror(E', h) → E''
  53. Mirror(C'_4, h) → N
  54. Polygon(G', E'', N) → poly2'', n, g'_1, e''_1
  55. Mirror(E'_1, h) → O
  56. Mirror(C''_2, h) → P
  57. Mirror(C'_3, h) → Q
  58. Polygon(O, P, Q) → poly1'', q, o, p
  59. Mirror(K, h) → K'
  60. Mirror(L, h) → L'
  61. Mirror(M, h) → M'
  62. Polygon(K', L', M') → poly4, m', k', l'
  63. Mirror(G'_1, h) → R
  64. Mirror(E''_1, h) → S
  65. Mirror(J, h) → J'
  66. Polygon(R, S, J') → poly5, j', r, s
  67. Mirror(C''_3, h) → T
  68. Mirror(D'''_1, h) → U
  69. Mirror(I'_1, h) → V
  70. Polygon(T, U, V) → poly6, v_1, t, u_1
  71. Vector((0, 0), (-3.27, 2.09)) → offsetvector
  72. Point(c) → W
  73. Rotate(W, 30*°, A) → W'
  74. Angle(W, A, W') → δ
  75. Vector(W', W) → z
  76. Vector(W', A) → w_1
  77. Polygon(TestPoint, E_1, D_1, J_1, TestB, Z, F_1, A_1, G_1, B_1, H_1, C_1, I_1) → poly7, testpoint, i_1, d_1, j_1, testb, e_3, z_1, f_1, a_1, g_2, b_1, h_1, c_4
  78. Translate(TestPoint, Vector[((g - 100) / 100 * w_1)]) → TestPoint'_1
  79. Translate(E_1, Vector[((g - 100) / 100 * w_1)]) → K_1
  80. Translate(Z, Vector[((g - 100) / 100 * w_1)]) → Z'_1
  81. Translate(F_1, Vector[((g - 100) / 100 * w_1)]) → L_1
  82. Translate(A_1, Vector[((g - 100) / 100 * w_1)]) → M_1
  83. Translate(G_1, Vector[((g - 100) / 100 * w_1)]) → N_1
  84. Translate(B_1, Vector[((g - 100) / 100 * w_1)]) → O_1
  85. Translate(H_1, Vector[((g - 100) / 100 * w_1)]) → P_1
  86. Translate(C_1, Vector[((g - 100) / 100 * w_1)]) → Q_1
  87. Translate(I_1, Vector[((g - 100) / 100 * w_1)]) → R_1
  88. Translate(D_1, Vector[((g - 100) / 100 * w_1)]) → S_1
  89. Translate(J_1, Vector[((g - 100) / 100 * w_1)]) → T_1
  90. Translate(TestB, Vector[((g - 100) / 100 * w_1)]) → TestB'_1
  91. Polygon(TestPoint'_1, K_1, S_1, T_1, TestB'_1, Z'_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1) → poly7'_1, testpoint', r_1, s_1, t_1, testb', k_1, z', l_1, m_1, n_1, o_1, p_1, q_1
  92. Mirror(TestPoint, h) → TestPoint'
  93. Mirror(E_1, h) → U_1
  94. Mirror(Z, h) → Z'
  95. Mirror(F_1, h) → V_1
  96. Mirror(A_1, h) → W_1
  97. Mirror(G_1, h) → Z_1
  98. Mirror(B_1, h) → A_2
  99. Mirror(H_1, h) → B_2
  100. Mirror(C_1, h) → C_2
  101. Mirror(I_1, h) → D_2
  102. Mirror(D_1, h) → E_2
  103. Mirror(J_1, h) → F_2
  104. Mirror(TestB, h) → TestB'
  105. Polygon(TestPoint', U_1, E_2, F_2, TestB', Z', V_1, W_1, Z_1, A_2, B_2, C_2, D_2) → poly7', testpoint'_1, d_2, e_4, f_2, testb'_1, u_2, z'_1, v_2, w_2, z_2, a_2, b_2, c_5
  106. Segment(G_2, H_2) → h_2
  107. Segment(I_2, J_2) → i_2

AI Drawing Prompt Reference

Create an animation demonstrating how to derive the area of a circle by approximating it with a parallelogram. Start by drawing a circle and a radius from the center to the circumference. Then, rotate and replicate this radius multiple times to form a series of parallelograms whose areas approach the circle's area. The animation should show the parallelograms rotating and reflecting to eventually form the circle, highlighting key geometric elements like the circle, radius, and angles to clearly illustrate the derivation process.
The prompt above was compiled from the original drawing commands and is for reference only.