两个正六边形重叠区域面积

两个正六边形重叠区域面积
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2012-12-17 11:29
①本范例演示如何计算两个正六边形重叠区域的面积;②涉及的核心数学概念包括多边形面积计算、几何作图、交集区域;③作图要点——从下方『构造摘要』提炼用到了 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/对象;④适合高中数学教学或几何问题探究,帮助学生理解多边形重叠区域的计算方法。

GeoGebra 命令及参数

  1. Polygon(B, A, 6) → poly1, b, F, a_1, c, d, e, f, C, D, E
  2. Segment(A, E) → g
  3. Segment(B, D) → h
  4. Intersect(g, h) → G
  5. Circle(G, b) → k
  6. Point(k) → H
  7. Polygon(H, G, 6) → poly2, h_1, L, g_1, i, j, l, m, I, J, K
  8. Intersect(b, g_1) → M
  9. Intersect(c, h_1) → N
  10. Polygon(A, M, G, N, C) → poly3, n, c_1, m_1, a_2, g_2
  11. Intersect(a_1, h_1) → O
  12. Intersect(f, g_1) → P
  13. Polygon(A, B, P, G, O) → poly4, a_3, b_1, p, g_3, o
  14. Intersect(b, h_1) → Q
  15. Intersect(e, g_1) → R
  16. Polygon(Q, B, F, R, G) → poly5, q, b_2, f_1, r, g_4
  17. Intersect(f, h_1) → S
  18. Intersect(d, g_1) → T
  19. Polygon(G, T, E, F, S) → poly6, g_5, t, e_1, f_2, s
  20. Intersect(c, g_1) → U
  21. Intersect(e, h_1) → V
  22. Polygon(U, D, E, V, G) → poly7, u, d_1, e_2, v, g_6
  23. Intersect(a_1, g_1) → W
  24. Intersect(h_1, d) → Z
  25. Polygon(W, C, D, Z, G) → poly8, w, c_2, d_2, z, g_7
  26. OrthogonalLine(G, b) → i_1
  27. Point(i_1) → A_1
  28. Segment(A_1, G) → j_1
  29. Angle(H, G, A_1) → α
  30. Segment(A, G) → a
  31. Segment(G, F) → k_1
  32. Segment(G, D) → l_1
  33. Circle(G, k_1) → p_1
  34. Point(p_1) → B_1
  35. Circle(B_1, f_2) → q_1
  36. Circle(G, s) → r_1
  37. Intersect(q_1, r_1, 2) → C_1
  38. Polygon(B_1, C_1, G) → poly9, g_8, b_3, c_3
  39. Circle(G, l_1) → s_1
  40. Point(s_1) → D_1
  41. Circle(D_1, u) → t_1
  42. Circle(G, g_6) → k_2
  43. Intersect(t_1, k_2, 1) → E_1
  44. Polygon(G, E_1, D_1) → poly10, d_3, g_9, e_3
  45. Circle(G, l_1) → p_2
  46. Point(p_2) → F_1
  47. Circle(F_1, d_2) → q_2
  48. Circle(G, z) → r_2
  49. Intersect(q_2, r_2, 2) → G_1
  50. Polygon(G, F_1, G_1) → poly11, g_{10}, g_{11}, f_3
  51. Circle(G, a) → s_2
  52. Segment(A, M) → n_1
  53. Point(s_2) → H_1
  54. Circle(H_1, n_1) → t_2
  55. Segment(M, G) → i_2
  56. Circle(G, i_2) → h_3
  57. Intersect(t_2, h_3, 1) → I_1
  58. Polygon(I_1, H_1, G) → poly12, g_{12}, i_3, h_4
  59. Circle(G, a) → k_3
  60. Point(k_3) → J_1
  61. Circle(J_1, o) → p_3
  62. Circle(G, g_3) → q_3
  63. Intersect(p_3, q_3, 2) → K_1
  64. Polygon(J_1, K_1, G) → poly13, g_{13}, j_2, k_4
  65. Circle(G, k_1) → r_3
  66. Circle(G, r) → s_3
  67. Point(r_3) → L_1
  68. Circle(L_1, f_1) → t_3
  69. Intersect(s_3, t_3, 2) → M_1
  70. Polygon(L_1, M_1, G) → poly14, g_{14}, l_2, m_2
  71. Polygon(G, T, E, F) → poly15, g_{15}, t_4, e_4, f_4
  72. Polygon(G, D, E, V) → poly16, g_{16}, d_4, e_5, v_1
  73. Polygon(G, W, C, D) → poly17, g_{17}, w_1, c_4, d_5
  74. Polygon(G, N, C, A) → poly18, g_{18}, n_2, c_5, a_4
  75. Polygon(G, A, B, P) → poly19, g_{19}, a_5, b_4, p_4
  76. Polygon(G, Q, B, F) → poly20, g_{20}, q_4, b_5, f_5
  77. PointIn(poly2) → N_1

AI 绘图提示语参考

画两个正六边形,并找出它们的重叠区域。首先绘制两个正六边形,然后找到它们的交点。以交点为中心,绘制一个圆,并在圆上找到一点。以这一点和交点为顶点,绘制一个新的正六边形。接着,分别以两个正六边形的边和交点为顶点,绘制多个多边形,形成重叠区域的边界。最后,测量并标注重叠区域的面积。
以上提示语为根据原始绘图命令整理而来,仅供参考。