Finding the Area of the Overlapped Region

Finding the Area of the Overlapped Region
Loading the math board and drawing, please wait…
2012-12-17 11:23
This example demonstrates how to calculate the area of the overlapped region between two squares. Key math concepts include area, rotation, square, perimeter, LCM and GCD. It uses GeoGebra commands like Polygon, Segment, Intersect, Circle, Point, OrthogonalLine, Angle, and PointIn for construction. Suitable for high school mathematics teaching, focusing on geometric calculations and problem-solving skills.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw two overlapping squares with one rotated relative to the other, and calculate the area of the overlapped region. Start by constructing the squares and their vertices, then find midpoints of the sides and connect them to form a new square. Use circles and intersection points to define the overlapped area, and subdivide it into smaller polygons. Finally, calculate the area of each polygon and sum them to find the total area of the overlap.
The prompt above was compiled from the original drawing commands and is for reference only.