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Area of Overlapped Regular Hexagons

Area of Overlapped Regular Hexagons
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2012-12-17 11:29
PolygonGeometric Construction
This example demonstrates how to calculate the area of the overlapped region between two regular hexagons. It involves core mathematical concepts such as polygon area calculation, geometric construction, and intersection regions. Key GeoGebra commands and objects used in the construction include Polygon, Segment, Intersect, Circle, Point, OrthogonalLine, Angle, and PointIn. Suitable for high school mathematics teaching or geometric problem exploration, helping students understand the calculation methods for overlapping polygonal areas.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw two regular hexagons and find their overlapping area. First, draw two regular hexagons and identify their intersection points. Center a circle at one of the intersection points and find a point on the circle. Use this point and the intersection point to draw a new regular hexagon. Then, draw multiple polygons using the sides of the hexagons and intersection points to form the boundaries of the overlapping region. Finally, measure and label the area of the overlapping region.
The prompt above was compiled from the original drawing commands and is for reference only.