求解重叠区域面积问题

求解重叠区域面积问题
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2012-12-17 11:23
①本范例演示两个正方形重叠区域的面积计算;②涉及核心数学概念包括面积、旋转、正方形、周长、最大公约数与最小公倍数;③作图要点运用了Polygon(多边形)、Segment(线段)、Intersect(交点)、Circle(圆)、Point(点)、OrthogonalLine(垂线)、Angle(角度)、PointIn(点在对象内)等GeoGebra命令;④适合高中数学教学,用于几何计算与问题解决训练。

GeoGebra 命令及参数

  1. Polygon(B, A, 4) → poly1, b, a_1, c, d, C, D
  2. Segment(A, D) → e
  3. Segment(B, C) → f
  4. Intersect(e, f) → E
  5. Circle(E, b) → g
  6. Point(g) → F
  7. Polygon(E, F, 4) → poly2, e_1, f_1, h, i, G, H
  8. Intersect(b, e_1) → I
  9. Intersect(a_1, i) → J
  10. Polygon(I, A, J, E) → poly3, i_1, a_2, j, e_2
  11. Intersect(b, i) → K
  12. Intersect(d, e_1) → L
  13. Polygon(L, B, K, E) → poly4, l, b_1, k, e_3
  14. Intersect(d, i) → M
  15. Intersect(c, e_1) → N
  16. Polygon(M, E, N, D) → poly5, m, e_4, n, d_1
  17. Intersect(c, i) → O
  18. Intersect(a_1, e_1) → P
  19. Polygon(E, O, C, P) → poly6, e_5, o, c_1, p
  20. OrthogonalLine(E, b) → a
  21. Point(a) → Q
  22. Segment(Q, E) → q
  23. Angle(H, E, Q) → α
  24. OrthogonalLine(E, a_1) → r
  25. Intersect(r, d) → R
  26. Segment(E, R) → s
  27. Segment(R, L) → t
  28. Segment(E, L) → g_1
  29. Circle(E, s) → h_1
  30. Point(h_1) → S
  31. Circle(S, t) → k_1
  32. Circle(E, g_1) → p_1
  33. Intersect(k_1, p_1, 1) → T
  34. Polygon(S, T, E) → poly7, e_6, s_1, t_1
  35. Circle(E, s) → q_1
  36. Segment(M, R) → j_1
  37. Segment(M, E) → l_1
  38. Point(q_1) → U
  39. Circle(E, l_1) → c_2
  40. Circle(U, j_1) → d_2
  41. Intersect(c_2, d_2, 1) → V
  42. Polygon(E, U, V) → poly9, v, e_8, u
  43. Intersect(a, c) → W
  44. Segment(E, N) → m_1
  45. Segment(W, N) → n_1
  46. Segment(E, W) → f_2
  47. Circle(E, m_1) → g_2
  48. Circle(E, f_2) → p_2
  49. Point(q_1) → X
  50. Circle(X, n_1) → h_2
  51. Circle(E, m_1) → q_2
  52. Intersect(h_2, q_2, 1) → Y
  53. Polygon(E, X, Y) → poly10, Z, e_9, A_1
  54. Intersect(r, a_1) → B_1
  55. Segment(E, B_1) → i_2
  56. Segment(B_1, P) → j_2
  57. Segment(P, E) → l_2
  58. Circle(E, l_2) → r_2
  59. Circle(E, i_2) → s_2
  60. Point(s_2) → C_1
  61. Circle(C_1, j_2) → t_2
  62. Intersect(r_2, t_2, 1) → D_1
  63. Polygon(E, C_1, D_1) → poly11, d_3, e_{10}, c_3
  64. Circle(E, i_2) → f_3
  65. Point(f_3) → E_1
  66. Segment(P, B_1) → m_2
  67. Circle(E_1, j_2) → g_3
  68. Circle(E, l_2) → h_3
  69. Intersect(g_3, h_3, 1) → F_1
  70. Polygon(E, F_1, E_1) → poly12, e_{11}, e_{12}, f_4
  71. Circle(E, i_2) → k_3
  72. Segment(E, J) → n_2
  73. Segment(B_1, J) → a_3
  74. Circle(E, n_2) → p_3
  75. Point(k_3) → G_1
  76. Circle(G_1, a_3) → q_3
  77. Intersect(p_3, q_3, 1) → H_1
  78. Polygon(E, G_1, H_1) → poly13, h_4, e_{13}, g_4
  79. Intersect(a, b) → I_1
  80. Segment(I_1, E) → b_3
  81. Segment(I_1, I) → i_3
  82. Segment(I, E) → j_3
  83. Circle(E, b_3) → r_3
  84. Point(r_3) → J_1
  85. Circle(E, j_3) → s_3
  86. Circle(J_1, i_3) → t_3
  87. Intersect(s_3, t_3, 2) → K_1
  88. Polygon(J_1, K_1, E) → poly14, e_{14}, j_4, k_4
  89. Segment(K, I_1) → l_3
  90. Segment(E, I_1) → m_3
  91. Segment(E, K) → n_3
  92. Circle(E, m_3) → c_4
  93. Circle(E, n_3) → d_4
  94. Point(c_4) → L_1
  95. Circle(L_1, l_3) → p_4
  96. Intersect(d_4, p_4, 1) → M_1
  97. Polygon(L_1, E, M_1) → poly15, m_4, l_4, e_{15}
  98. Segment(R, E) → a_4
  99. Polygon(B, L, E, I_1) → poly8, r_1, b_2, k_2, b_4
  100. Polygon(K, B, R, E) → poly16, k_5, b_5, r_4, e_7
  101. Polygon(N, D, R, E) → poly17, n_4, d_5, r_5, e_{16}
  102. Polygon(W, D, M, E) → poly18, w, d_6, m_5, e_{17}
  103. Polygon(W, C, P, E) → poly19, w_1, c_5, p_5, e_{18}
  104. Polygon(O, C, B_1, E) → poly20, o_1, c_6, b_6, e_{19}
  105. Polygon(I, A, B_1, E) → poly21, i_4, a_5, b_7, e_{20}
  106. Polygon(J, A, I_1, E) → poly22, j_5, a_6, i_5, e_{21}
  107. PointIn(poly2) → N_1

AI 绘图提示语参考

画两个正方形,其中一个旋转一定角度与另一个重叠,求重叠部分的面积。先画出正方形的边和顶点,然后找到边的中点并连接,形成新的正方形。接着,利用圆和交点构造重叠区域,并细分重叠部分为多个小多边形,最后计算每个小多边形的面积并求和得到总面积。
以上提示语为根据原始绘图命令整理而来,仅供参考。