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Circle Area Calculation by GeoGebra Materials Team

Circle Area Calculation by GeoGebra Materials Team
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2012-02-22 19:47
CircleTrigonometryCoordinate GeometryOptimization
In collections Area, Surface Area, Volume--Nets
This example demonstrates how to calculate the area of a circle by dividing it into sectors and transforming them into triangles. Key math concepts include circle, trigonometry, coordinate geometry, and optimization. Construction highlights involve GeoGebra commands like Circle, Point, Rotate, Polygon, Line, OrthogonalLine, Segment, Midpoint, Intersect, Vector, Translate, If, and Mirror. Suitable for high school mathematics education to understand the derivation of the circle area formula.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Rotate(C, 36*°, A) → D
  4. Rotate(D, 36*°, A) → E
  5. Rotate(E, 36*°, A) → F
  6. Rotate(F, 36*°, A) → G
  7. Rotate(G, 36*°, A) → H
  8. Rotate(H, 36*°, A) → I
  9. Rotate(I, 36*°, A) → J
  10. Rotate(J, 36*°, A) → K
  11. Rotate(K, 36*°, A) → L
  12. Rotate(L, 36*°, A) → M
  13. Polygon(M, D, A) → P, a, m, d
  14. Polygon(D, E, A) → Q, a_1, d_1, e
  15. Polygon(E, F, A) → R, a_2, e_1, f
  16. Polygon(F, G, A) → S, a_3, f_1, g
  17. Polygon(G, H, A) → T, a_4, g_1, h
  18. Line(C, A) → b
  19. OrthogonalLine(C, b) → n
  20. Segment(E, F) → p
  21. Midpoint(E, F) → N
  22. Circle(C, p) → q
  23. Intersect(q, n, 2) → O
  24. Intersect(q, n, 1) → A_1
  25. Vector(C, O) → u
  26. Translate(O, u) → B_1
  27. Translate(B_1, u) → C_1
  28. Translate(C_1, u) → D_1
  29. Translate(D_1, u) → E_1
  30. Rotate(D, -r, M) → D'
  31. Rotate(E, -r, M) → E'
  32. Rotate(A, -r, M) → A'_1
  33. Rotate(F, -r, M) → F'
  34. Rotate(G, -r, M) → G'
  35. Rotate(H, -r, M) → H'
  36. Rotate(A, 342*°, M) → F_1
  37. Rotate(O, 36*°, F_1) → G_1
  38. Rotate(G_1, 36*°, F_1) → H_1
  39. Rotate(H_1, 36*°, F_1) → I_1
  40. Rotate(I_1, 36*°, F_1) → J_1
  41. Rotate(F_1, -(r - 18°), O) → F1
  42. Rotate(G_1, -(r - 18°), O) → G1
  43. Rotate(H_1, -(r - 18°), O) → H1
  44. Rotate(I_1, -(r - 18°), O) → I2
  45. Rotate(J_1, -(r - 18°), O) → J2
  46. Translate(F_1, u) → K_1
  47. Translate(K_1, u) → L_1
  48. Translate(L_1, u) → M_1
  49. Translate(M_1, u) → N_1
  50. Rotate(B_1, 36*°, K_1) → O_1
  51. Rotate(O_1, 36*°, K_1) → T_1
  52. Rotate(T_1, 36*°, K_1) → U_1
  53. Rotate(K_1, -(r - 54°), B_1) → K1
  54. Rotate(O_1, -(r - 54°), B_1) → O1
  55. Rotate(T_1, -(r - 54°), B_1) → T2
  56. Rotate(U_1, -(r - 54°), B_1) → U2
  57. Rotate(C_1, 36*°, L_1) → Y_1
  58. Rotate(Y_1, 36*°, L_1) → Z_1
  59. Rotate(Y_1, -(r - 90°), C_1) → Y1
  60. Rotate(L_1, -(r - 90°), C_1) → L2
  61. Rotate(Z_1, -(r - 90°), C_1) → Z1
  62. Rotate(D_1, 36*°, M_1) → A_2
  63. Rotate(A_2, -(r - 126°), D_1) → B_2
  64. Rotate(M_1, -(r - 126°), D_1) → C_2
  65. Polygon(E_1, N_1, M, D_1, M_1, C_1, L_1, B_1, K_1, O, F_1) → P_1, e_2, f_2, m_3, n_1, d_2, m_2, c_1, l_2, b_1, k_2, o
  66. Midpoint(B_1, L_1) → Q_1
  67. Rotate(E_1, -(r - 162°), Q_1) → E3
  68. Rotate(N_1, -(r - 162°), Q_1) → N_1'
  69. Rotate(D_1, -(r - 162°), Q_1) → D_1'
  70. Rotate(M_1, -(r - 162°), Q_1) → M5
  71. Rotate(C_1, -(r - 162°), Q_1) → C_1'
  72. Rotate(L_1, -(r - 162°), Q_1) → L_2
  73. Rotate(B_1, -(r - 162°), Q_1) → B_1'
  74. Rotate(K_1, -(r - 162°), Q_1) → K_2
  75. Rotate(O, -(r - 162°), Q_1) → O'
  76. Rotate(F_1, -(r - 162°), Q_1) → F_2
  77. Rotate(M, -(r - 162°), Q_1) → M'
  78. If(r ≥ 162°, M') → R_1
  79. Polygon(R_1, F_2, E3, O', K_2, B_1', L_2, C_1', M5, D_1', N_1') → S_1, r_1, n_1', e_1', f_3, o', k_3, b_1', l_3, c_1', m_1', d_1'
  80. If(r ≤ 18°, D', O) → V_1
  81. If(r ≤ 18°, A'_1, F_1) → W_1
  82. Polygon(V_1, W_1, M) → P_2, m_4, v_1, w_1
  83. If(r ≤ 18°, E', If[r ≤ 54°, G1, B_1]) → D_2
  84. If(r ≤ 18°, W_1, If[r ≤ 54°, F1, K_1]) → E_2
  85. Polygon(D_2, E_2, V_1) → Q_2, v_2, d_3, e_3
  86. If(r ≤ 18°, F', If[r ≤ 54°, H1, If[r ≤ 90°, O1, C_1]]) → G_2
  87. If(r ≤ 54°, E_2, If[r ≤ 90°, K1, L_1]) → H_2
  88. Polygon(G_2, H_2, D_2) → R_2, d_4, g_2, h_2
  89. If(r ≤ 90°, H_2, If[r ≤ 126°, L2, M_1]) → I_2
  90. If(r ≤ 126°, I_2, If[r ≤ 162°, C_2, N_1]) → J_2
  91. If(r ≤ 18°, G', If[r ≤ 54°, I2, If[r ≤ 90°, T2, If[r ≤ 126°, Y1, D_1]]]) → M_2
  92. Polygon(M_2, I_2, G_2) → S_2, g_3, m_5, i_2
  93. Rotate(H, -r, M) → N_2
  94. If(r ≤ 18°, N_2, If[r ≤ 54°, J2, If[r ≤ 90°, U2, If[r ≤ 126°, Z1, If[r ≤ 162°, B_2, E_1]]]]) → O_2
  95. Polygon(O_2, J_2, M_2) → T_2, m_6, o_2, j_2
  96. If(r ≟ 342°, O_2) → U_2
  97. If(r ≟ 342°, J_2) → V_2
  98. If(r ≟ 342°, E3) → W_2
  99. If(r ≟ 342°, M) → Y_2
  100. Mirror(V_1, b) → V6
  101. Mirror(W_1, b) → W_1'
  102. Mirror(M, b) → M6
  103. Polygon(V6, W_1', M6) → P_2', m'_1, v_1', w_1'
  104. Mirror(D_2, b) → D7
  105. Mirror(E_2, b) → E_2'
  106. Mirror(V_1, b) → V_3
  107. Polygon(D7, E_2', V_3) → Q_2', v_4, d_2', e_2'
  108. Mirror(G_2, b) → G6
  109. Mirror(H_2, b) → H_2'
  110. Mirror(D_2, b) → D_3
  111. Polygon(G6, H_2', D_3) → R_2', d_5, g_2', h_2'
  112. Mirror(M_2, b) → M4
  113. Mirror(I_2, b) → I_2'
  114. Mirror(G_2, b) → G_3
  115. Polygon(M4, I_2', G_3) → S_2', g_4, m_2', i_2'
  116. Mirror(O_2, b) → O4
  117. Mirror(J_2, b) → J_2'
  118. Mirror(M_2, b) → M_3
  119. Polygon(O4, J_2', M_3) → T_2', m_1, o_2', j_2'
  120. If(r ≥ 162°, O4) → U
  121. If(r ≥ 162°, M4) → V
  122. If(r ≥ 162°, G6) → W
  123. If(r ≥ 162°, D7) → Y
  124. If(r ≥ 162°, V6) → Z
  125. If(r ≥ 162°, M6) → A_3
  126. Polygon(U, J_2', A_3, V, I_2', W, H_2', Y, E_2', Z, W_1') → P_3, u_1, w_3, a_5, j_1, v, i_1, w, h_1, y, e_4, z
  127. Mirror(C_1, b) → B_3
  128. Mirror(L_1, b) → C_3
  129. Midpoint(B_3, C_3) → E_3
  130. Rotate(U, -(r - 162°), E_3) → U'
  131. Rotate(J_2', -(r - 162°), E_3) → J_2''
  132. Rotate(V, -(r - 162°), E_3) → V'
  133. Rotate(I_2', -(r - 162°), E_3) → I_2''
  134. Rotate(W, -(r - 162°), E_3) → W'
  135. Rotate(H_2', -(r - 162°), E_3) → H_2''
  136. Rotate(Y, -(r - 162°), E_3) → Y'
  137. Rotate(E_2', -(r - 162°), E_3) → E_2''
  138. Rotate(Z, -(r - 162°), E_3) → Z'
  139. Rotate(W_1', -(r - 162°), E_3) → W9
  140. Rotate(A_3, -(r - 162°), E_3) → A8
  141. Polygon(U', J_2'', A8, V', I_2'', W', H_2'', Y', E_2'', Z', W9) → P_3', u', w_1'', a_3', j_2'', v', i_2'', w', h_2'', y', e_2'', z'
  142. If(r ≟ 342°, W9) → Z_2
  143. If(r ≟ 342°, A8) → F_3
  144. Polygon(U_2, Z_2, F_3, V_2) → Q_3, u_2, z_2, f_4, v_3
  145. If(r ≟ 0°, C) → H_3
  146. If(r ≟ 0°, A) → I_3
  147. Circle(I_3, H_3) → k
  148. If(Hint ≟ 1, C) → J_3
  149. Line(L_1, n) → i
  150. Point(n) → K_3
  151. OrthogonalLine(K_3, n) → j
  152. Intersect(i, j) → L_3
  153. Midpoint(K_3, L_3) → N_3
  154. Point(j) → O_3
  155. Midpoint(O_3, K_3) → R_3
  156. If(Hint ≟ 1, O_3) → S_3
  157. If(Hint ≟ 1, R_3) → T_3
  158. If(Hint ≟ 1, K_3) → U_3
  159. Segment(U_3, S_3) → l
  160. Mirror(U_3, b) → U_3'
  161. Mirror(S_3, b) → S_3'
  162. Segment(U_3', S_3') → s
  163. Mirror(T_3, b) → W_3
  164. If(Hint ≟ 1, O) → Y_3
  165. OrthogonalLine(Y_3, i) → t
  166. OrthogonalLine(S_3', t) → k_1
  167. Intersect(t, k_1) → Z_3
  168. Midpoint(Z_3, Y_3) → A_4
  169. Vector(A_4, W_3) → z_1
  170. Mirror(A_4, b) → C_4
  171. Vector(C_4, T_3) → l_1
  172. Line(N_3, i) → p_1
  173. Point(p_1) → B_4
  174. If(Hint ≟ 1, B_4) → D_4
  175. OrthogonalLine(D_4, p_1) → q_1
  176. Intersect(n, q_1) → E_4
  177. Intersect(q_1, i) → F_4
  178. Point(n) → G_4
  179. Segment(E_4, G_4) → s_1
  180. OrthogonalLine(G_4, i) → t_1
  181. Intersect(i, t_1) → H_4
  182. Segment(F_4, H_4) → b_2
  183. Midpoint(E_4, G_4) → I_4
  184. Midpoint(F_4, H_4) → J_4
  185. OrthogonalLine(I_4, i) → c_2
  186. Point(c_2) → K_4
  187. Vector(K_4, I_4) → w_2
  188. Point(c_2) → L_4
  189. Point(c_2) → M_4
  190. Vector(M_4, I_4) → p_2
  191. Point(c_2) → N_4
  192. Vector(N_4, J_4) → q_2