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Area of a Circle: Geometry-Based Construction and Illustration

Area of a Circle: Geometry-Based Construction and Illustration
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2023-03-13 04:12
CircleVectorIntegralTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to build an intuitive, geometry-centered visualization of the area of a circle using constructed points and symmetric transformations. It focuses on core ideas such as the circle (radius/center), area as a measurable quantity, and how rotations and reflections can generate structured regions that support a geometric partition/approximation viewpoint. Construction highlights in GeoGebra include Midpoint (to create key reference points), Circle (to draw the main circle), Rotate and Mirror (to enforce rotational and reflection symmetry), Line and Polygon (to form connected partition boundaries), and Vector (to show direction/associated measures). An If statement is used to control conditional construction steps, while angle/auxiliary parameters (e.g., α) and many generated points (such as P_i, A_i, E_i) are used to systematically assemble the final symmetric segmented layout. Suitable for secondary school through early university math instruction, especially for learners exploring geometric construction, measurement, and the intuition leading toward integral-based area reasoning.

GeoGebra Commands & Parameters

  1. Midpoint(N_1, O_1) → V_1
  2. Midpoint(L_1, M_1) → S_1
  3. Circle(A, P_0) → c
  4. Rotate(P_0, α, A) → A_1
  5. If(a < 1, a, 1) → d
  6. Rotate(A_1, ((-d) * α / 2), P_0) → P_1
  7. Rotate(A, ((-d) * α / 2), P_0) → E_1
  8. Rotate(P_1, α, E_1) → A_2
  9. Rotate(A_2, ((-d) * α), P_1) → P_2
  10. Rotate(E_1, ((-d) * α), P_1) → E_2
  11. Rotate(P_2, α, E_2) → A_3
  12. Rotate(A_3, ((-d) * α), P_2) → P_3
  13. Rotate(E_2, ((-d) * α), P_2) → E_3
  14. Rotate(P_3, α, E_3) → A_4
  15. Rotate(A_4, ((-d) * α), P_3) → P_4
  16. Rotate(E_3, ((-d) * α), P_3) → E_4
  17. Rotate(E_4, ((-d) * α), P_4) → E_5
  18. Rotate(P_4, α, E_4) → A_5
  19. Rotate(A_5, ((-d) * α), P_4) → P_5
  20. Rotate(E_5, ((-d) * α), P_5) → E_6
  21. Rotate(P_5, α, E_5) → A_6
  22. Rotate(A_6, ((-d) * α), P_5) → P_6
  23. Rotate(E_6, ((-d) * α), P_6) → E_7
  24. Rotate(P_6, α, E_6) → A_7
  25. Rotate(A_7, ((-d) * α), P_6) → P_7
  26. Rotate(E_7, ((-d) * α), P_7) → E_8
  27. Rotate(P_7, α, E_7) → A_8
  28. Rotate(A_8, ((-d) * α), P_7) → P_8
  29. Line(P_0, A) → b
  30. Rotate(E_8, ((-d) * α), P_8) → E_9
  31. Rotate(P_8, α, E_8) → A_9
  32. Rotate(A_9, ((-d) * α), P_8) → P_9
  33. Rotate(E_9, ((-d) * α), P_9) → F_0
  34. Rotate(P_9, α, E_9) → B_0
  35. Rotate(B_0, ((-d) * α), P_9) → Q_0
  36. Rotate(F_0, ((-d) * α), Q_0) → F_1
  37. Rotate(Q_0, α, F_0) → B_1
  38. Rotate(B_1, ((-d) * α), Q_0) → Q_1
  39. Rotate(F_1, ((-d) * α), Q_1) → F_2
  40. Rotate(Q_1, α, F_1) → B_2
  41. Rotate(B_2, ((-d) * α), Q_1) → Q_2
  42. Rotate(F_2, ((-d) * α), Q_2) → F_3
  43. Rotate(Q_2, α, F_2) → B_3
  44. Rotate(B_3, ((-d) * α), Q_2) → Q_3
  45. Rotate(F_3, ((-d) * α), Q_3) → F_4
  46. Rotate(Q_3, α, F_3) → B_4
  47. Rotate(B_4, ((-d) * α), Q_3) → Q_4
  48. Rotate(F_4, ((-d) * α), Q_4) → F_5
  49. Rotate(Q_4, α, F_4) → B_5
  50. Rotate(B_5, ((-d) * α), Q_4) → Q_5
  51. Mirror(E_1, b) → B
  52. Mirror(E_2, b) → C
  53. Mirror(E_3, b) → D
  54. Mirror(E_4, b) → E
  55. Mirror(E_6, b) → F
  56. Mirror(E_5, b) → G
  57. Mirror(E_7, b) → H
  58. Mirror(E_8, b) → I
  59. Mirror(E_9, b) → J
  60. Mirror(F_0, b) → K
  61. Mirror(F_2, b) → L
  62. Mirror(F_3, b) → N
  63. Mirror(F_4, b) → O
  64. Mirror(F_5, b) → P
  65. Mirror(F_1, b) → M
  66. Mirror(P_1, b) → Q
  67. Mirror(P_2, b) → R
  68. Mirror(P_3, b) → S
  69. Mirror(P_4, b) → T
  70. Mirror(P_5, b) → U
  71. Mirror(P_6, b) → V
  72. Mirror(P_7, b) → W
  73. Mirror(P_8, b) → Y
  74. Mirror(P_9, b) → Z
  75. Mirror(Q_1, b) → C_1
  76. Mirror(Q_0, b) → D_1
  77. Mirror(Q_2, b) → G_1
  78. Mirror(Q_3, b) → H_1
  79. Mirror(Q_4, b) → I_1
  80. Mirror(Q_5, b) → J_1
  81. Polygon(F_5, Q_5, P_6, P_5, P_4, P_3, P_2, P_1, P_0, Q, R, S, Q_4, T, U, V, W, Y, Z, D_1, C_1, G_1, H_1, Q_3, I_1, J_1, P, I_1, O, H_1, N, G_1, L, C_1, Q_2, M, D_1, K, Z, J, Y, I, W, H, V, Q_1, F, U, G, T, E, S, D, R, C, Q, Q_0, B, P_0, E_1, P_1, E_2, P_2, E_3, P_3, E_4, P_4, P_9, E_5, P_5, E_6, P_6, E_7, P_7, E_8, P_8, E_9, P_9, P_8, F_0, Q_0, F_1, Q_1, F_2, Q_2, F_3, Q_3, F_4, Q_4, P_7) → R_1, e, n, p, q, r, s, t, a_1, b_1, c_1, d_1, f, e_1, f_1, g_1, h_1, i_1, j_1, k_1, l_1, m_1, n_1, g, p_1, q_1, r_1, s_1, t_1, a_2, b_2, c_2, d_2, e_2, h, f_2, g_2, h_2, i_2, j_2, k_2, l_2, m_2, n_2, p_2, i, q_2, r_2, s_2, t_2, a_3, b_3, c_3, d_3, e_3, f_3, j, g_3, h_3, i_3, j_3, k_3, l_3, m_3, n_3, p_3, q_3, k, r_3, s_3, t_3, a_4, b_4, c_4, d_4, e_4, f_4, g_4, l, h_4, i_4, j_4, k_4, l_4, m_4, n_4, p_4, q_4, r_4, m, s_4
  82. If(a < 1.2, 0, a - 1.2) → dd
  83. Midpoint(P_0, B) → K_1
  84. Rotate(F_5, (dd * 3.141592653589793), K_1) → F_5'
  85. Rotate(Q_5, (dd * 3.141592653589793), K_1) → Q_5'
  86. Rotate(Q_4, (dd * 3.141592653589793), K_1) → Q_4'
  87. Rotate(Q_3, (dd * 3.141592653589793), K_1) → Q_3'
  88. Rotate(Q_2, (dd * 3.141592653589793), K_1) → Q_2'
  89. Rotate(Q_1, (dd * 3.141592653589793), K_1) → Q_1'
  90. Rotate(Q_0, (dd * 3.141592653589793), K_1) → Q_0'
  91. Rotate(P_9, (dd * 3.141592653589793), K_1) → P_9'
  92. Rotate(P_8, (dd * 3.141592653589793), K_1) → P_8'
  93. Rotate(P_7, (dd * 3.141592653589793), K_1) → P_7'
  94. Rotate(P_6, (dd * 3.141592653589793), K_1) → P_6'
  95. Rotate(P_5, (dd * 3.141592653589793), K_1) → P_5'
  96. Rotate(P_4, (dd * 3.141592653589793), K_1) → P_4'
  97. Rotate(P_3, (dd * 3.141592653589793), K_1) → P_3'
  98. Rotate(P_2, (dd * 3.141592653589793), K_1) → P_2'
  99. Rotate(P_1, (dd * 3.141592653589793), K_1) → P_1'
  100. Rotate(P_0, (dd * 3.141592653589793), K_1) → P_0'
  101. Rotate(Q, (dd * 3.141592653589793), K_1) → Q'
  102. Rotate(R, (dd * 3.141592653589793), K_1) → R'
  103. Rotate(S, (dd * 3.141592653589793), K_1) → S'
  104. Rotate(T, (dd * 3.141592653589793), K_1) → T'
  105. Rotate(U, (dd * 3.141592653589793), K_1) → U'
  106. Rotate(V, (dd * 3.141592653589793), K_1) → V'
  107. Rotate(W, (dd * 3.141592653589793), K_1) → W'
  108. Rotate(Y, (dd * 3.141592653589793), K_1) → Y'
  109. Rotate(Z, (dd * 3.141592653589793), K_1) → Z'
  110. Rotate(D_1, (dd * 3.141592653589793), K_1) → D_1'
  111. Rotate(C_1, (dd * 3.141592653589793), K_1) → C_1'
  112. Rotate(G_1, (dd * 3.141592653589793), K_1) → G_1'
  113. Rotate(H_1, (dd * 3.141592653589793), K_1) → H_1'
  114. Rotate(I_1, (dd * 3.141592653589793), K_1) → I_1'
  115. Rotate(J_1, (dd * 3.141592653589793), K_1) → J_1'
  116. Rotate(P, (dd * 3.141592653589793), K_1) → P'
  117. Rotate(I_1, (dd * 3.141592653589793), K_1) → I_2
  118. Rotate(O, (dd * 3.141592653589793), K_1) → O'
  119. Rotate(H_1, (dd * 3.141592653589793), K_1) → H_2
  120. Rotate(N, (dd * 3.141592653589793), K_1) → N'
  121. Rotate(G_1, (dd * 3.141592653589793), K_1) → G_2
  122. Rotate(L, (dd * 3.141592653589793), K_1) → L'
  123. Rotate(C_1, (dd * 3.141592653589793), K_1) → C_2
  124. Rotate(M, (dd * 3.141592653589793), K_1) → M'
  125. Rotate(D_1, (dd * 3.141592653589793), K_1) → D_2
  126. Rotate(K, (dd * 3.141592653589793), K_1) → K'
  127. Rotate(Z, (dd * 3.141592653589793), K_1) → Z'_1
  128. Rotate(J, (dd * 3.141592653589793), K_1) → J'
  129. Rotate(Y, (dd * 3.141592653589793), K_1) → Y'_1
  130. Rotate(I, (dd * 3.141592653589793), K_1) → I'
  131. Rotate(W, (dd * 3.141592653589793), K_1) → W'_1
  132. Rotate(H, (dd * 3.141592653589793), K_1) → H'
  133. Rotate(V, (dd * 3.141592653589793), K_1) → V'_1
  134. Rotate(F, (dd * 3.141592653589793), K_1) → F'
  135. Rotate(U, (dd * 3.141592653589793), K_1) → U'_1
  136. Rotate(G, (dd * 3.141592653589793), K_1) → G'
  137. Rotate(T, (dd * 3.141592653589793), K_1) → T'_1
  138. Rotate(E, (dd * 3.141592653589793), K_1) → E'
  139. Rotate(S, (dd * 3.141592653589793), K_1) → S'_1
  140. Rotate(D, (dd * 3.141592653589793), K_1) → D'
  141. Rotate(R, (dd * 3.141592653589793), K_1) → R'_1
  142. Rotate(C, (dd * 3.141592653589793), K_1) → C'
  143. Rotate(Q, (dd * 3.141592653589793), K_1) → Q'_1
  144. Rotate(B, (dd * 3.141592653589793), K_1) → B'
  145. Rotate(P_0, (dd * 3.141592653589793), K_1) → P_{10}
  146. Rotate(E_1, (dd * 3.141592653589793), K_1) → E_1'
  147. Rotate(P_1, (dd * 3.141592653589793), K_1) → P_{11}
  148. Rotate(E_2, (dd * 3.141592653589793), K_1) → E_2'
  149. Rotate(P_2, (dd * 3.141592653589793), K_1) → P_{12}
  150. Rotate(E_3, (dd * 3.141592653589793), K_1) → E_3'
  151. Rotate(P_3, (dd * 3.141592653589793), K_1) → P_{13}
  152. Rotate(E_4, (dd * 3.141592653589793), K_1) → E_4'
  153. Rotate(P_4, (dd * 3.141592653589793), K_1) → P_{14}
  154. Rotate(E_5, (dd * 3.141592653589793), K_1) → E_5'
  155. Rotate(P_5, (dd * 3.141592653589793), K_1) → P_{15}
  156. Rotate(E_6, (dd * 3.141592653589793), K_1) → E_6'
  157. Rotate(P_6, (dd * 3.141592653589793), K_1) → P_{16}
  158. Rotate(E_7, (dd * 3.141592653589793), K_1) → E_7'
  159. Rotate(P_7, (dd * 3.141592653589793), K_1) → P_{17}
  160. Rotate(E_8, (dd * 3.141592653589793), K_1) → E_8'
  161. Rotate(P_8, (dd * 3.141592653589793), K_1) → P_{18}
  162. Rotate(E_9, (dd * 3.141592653589793), K_1) → E_9'
  163. Rotate(P_9, (dd * 3.141592653589793), K_1) → P_{19}
  164. Rotate(F_0, (dd * 3.141592653589793), K_1) → F_0'
  165. Rotate(Q_0, (dd * 3.141592653589793), K_1) → Q_6
  166. Rotate(F_1, (dd * 3.141592653589793), K_1) → F_1'
  167. Rotate(Q_1, (dd * 3.141592653589793), K_1) → Q_7
  168. Rotate(F_2, (dd * 3.141592653589793), K_1) → F_2'
  169. Rotate(Q_2, (dd * 3.141592653589793), K_1) → Q_8
  170. Rotate(F_3, (dd * 3.141592653589793), K_1) → F_3'
  171. Rotate(Q_3, (dd * 3.141592653589793), K_1) → Q_9
  172. Rotate(F_4, (dd * 3.141592653589793), K_1) → F_4'
  173. Rotate(Q_4, (dd * 3.141592653589793), K_1) → Q_{10}
  174. Polygon(F_5', Q_5', P_6', P_5', P_4', P_3', P_2', P_1', P_0', Q', R', S', Q_4', T', U', V', W', Y', Z', D_1', C_1', G_1', H_1', Q_3', I_1', J_1', P', I_2, O', H_2, N', G_2, L', C_2, Q_2', M', D_2, K', Z'_1, J', Y'_1, I', W'_1, H', V'_1, Q_1', F', U'_1, G', T'_1, E', S'_1, D', R'_1, C', Q'_1, Q_0', B', P_{10}, E_1', P_{11}, E_2', P_{12}, E_3', P_{13}, E_4', P_{14}, P_9', E_5', P_{15}, E_6', P_{16}, E_7', P_{17}, E_8', P_{18}, E_9', P_{19}, P_8', F_0', Q_6, F_1', Q_7, F_2', Q_8, F_3', Q_9, F_4', Q_{10}, P_7') → R_1', f_5', p_7', p_6', p_5', p_4', p_3', p_2', p_1', p_0', q', r', q_5', s', t', u', v', w', y', z', d_1', c_1', g_1', q_4', h_1', i_1', j_1', p', i_5, o', h_5, n', g_5, l', q_3', c_5, m', d_5, k', z'_1, j', y'_1, i', w'_1, h', q_2', v'_1, f', u'_1, g', t'_1, e', s'_1, d', r'_1, c', q_1', q'_1, b', p_{10}, e_1', p_{11}, e_2', p_{12}, e_3', p_{13}, e_4', q_0', p_{14}, e_5', p_{15}, e_6', p_{16}, e_7', p_{17}, e_8', p_{18}, e_9', p_9', p_{19}, f_0', q_6, f_1', q_7, f_2', q_8, f_3', q_9, f_4', p_8', q_{10}
  175. Vector(S_1, T_1) → v
  176. Vector(S_1, U_1) → u
  177. Vector(V_1, W_1) → w
  178. Vector(V_1, Y_1) → z

AI Drawing Prompt Reference

Draw a graphic consisting of a circle and a series of symmetric line segments. The circle has its center at the origin and a unit radius. Through a series of rotations and reflections, multiple symmetric points are generated, which are used to connect positions on the circumference. The final graphic is formed by the circle and line segments connecting these points, creating symmetric sectors or regions. The construction order is: first draw the circle, then generate symmetric points via rotations and reflections, and finally connect these points with line segments to form the overall pattern. The graphic should maintain symmetry and reflect the intuitive concept of geometric partitioning.
The prompt above was compiled from the original drawing commands and is for reference only.