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Choosing Suitable Shapes for Measuring Areas

Choosing Suitable Shapes for Measuring Areas
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2014-12-09 12:33
CirclePolygonGeometric Construction
In collections P4 Area (I)
This example demonstrates how to choose suitable shapes for measuring areas. It involves core geometric concepts such as polygons, circles, and triangles. Using GeoGebra commands like Sequence, Polygon, Midpoint, etc., students can understand how to construct and analyze geometric shapes. Suitable for middle school and above students learning geometry and area measurement.

GeoGebra Commands & Parameters

  1. Sequence(E + (i, 0), i, 0.5, 3, 0.1) → list2
  2. Polygon((0, 0), (6, 0), (6, 4), (0, 4)) → poly2, e, f, g, h
  3. Polygon((8, 0), (13, 0), (13, 5), (8, 5)) → poly1, a, b, c, d
  4. Midpoint(d) → A
  5. Midpoint(a) → B
  6. Midpoint(f) → C
  7. Midpoint(e) → D
  8. Circle((3.75, 6.75), 0.75) → k
  9. Circle((14.25, 8.25), 0.75) → p_{10}
  10. Circle((14.25, 9.75), 0.75) → q_{10}
  11. Circle((12.75, 6.75), 0.75) → r_{10}
  12. Circle((9.75, 6.75), 0.75) → s_{10}
  13. Circle((8.25, 6.75), 0.75) → t_{10}
  14. Circle((11.25, 6.75), 0.75) → k_{28}
  15. Circle((5.25, 6.75), 0.75) → p_{28}
  16. Circle((6.75, 6.75), 0.75) → q_{28}
  17. Circle((0.75, 6.75), 0.75) → r_{28}
  18. Circle((2.25, 6.779603085772247), 0.75) → s_{28}
  19. Circle((12.75, 8.25), 0.75) → t_{28}
  20. Circle((11.25, 8.25), 0.75) → c_{29}
  21. Circle((9.75, 8.25), 0.75) → d_{29}
  22. Circle((8.25, 8.25), 0.75) → e_{29}
  23. Circle((6.75, 8.25), 0.75) → f_{29}
  24. Circle((5.25, 8.25), 0.75) → g_{29}
  25. Circle((3.75, 8.25), 0.75) → h_{29}
  26. Circle((2.25, 8.25), 0.75) → k_{29}
  27. Circle((3.75, 9.75), 0.75) → p_{29}
  28. Circle((2.25, 9.75), 0.75) → q_{29}
  29. Circle((14.25, 6.75), 0.75) → r_{29}
  30. Circle((0.75, 8.25), 0.75) → s_{29}
  31. Circle((0.75, 9.75), 0.75) → t_{29}
  32. Circle((5.25, 9.75), 0.75) → c_{30}
  33. Circle((6.75, 9.75), 0.75) → d_{30}
  34. Circle((8.25, 9.75), 0.75) → e_{30}
  35. Circle((9.75, 9.75), 0.75) → f_{30}
  36. Circle((11.25, 9.75), 0.75) → g_{30}
  37. Circle((12.75, 9.75), 0.75) → h_{30}
  38. Polygon((15, 10), (18, 10), (18, 12), (15, 12)) → poly4, m_{10}, n_{10}, a_{11}, l_{28}
  39. Polygon((12, 10), (15, 10), (15, 12), (12, 12)) → poly5, e_{10}, f_{10}, g_{10}, h_{10}
  40. Polygon((9, 10), (12, 10), (12, 12), (9, 12)) → poly6, i_{10}, j_{10}, k_{10}, l_{10}
  41. Polygon((6, 10), (9, 10), (9, 12), (6, 12)) → poly7, j_{11}, k_{11}, l_{11}, m_{11}
  42. Polygon((3, 10), (6, 10), (6, 12), (3, 12)) → poly8, n_{11}, p_{11}, q_{11}, r_{11}
  43. Polygon((0, 10), (3, 10), (3, 12), (0, 12)) → poly9, s_{11}, t_{11}, a_{12}, b_{12}
  44. Polygon((15, 8), (18, 8), (18, 10), (15, 10)) → poly10, c_{12}, d_{12}, e_{12}, f_{12}
  45. Polygon((12, 8), (15, 8), (15, 10), (12, 10)) → poly11, g_{12}, h_{12}, i_{12}, j_{12}
  46. Polygon((9, 8), (12, 8), (12, 10), (9, 10)) → poly12, k_{12}, l_{12}, m_{12}, n_{12}
  47. Polygon((6, 8), (9, 8), (9, 10), (6, 10)) → poly13, p_{12}, q_{12}, r_{12}, s_{12}
  48. Polygon((3, 8), (6, 8), (6, 10), (3, 10)) → poly14, t_{12}, a_{13}, b_{13}, c_{13}
  49. Polygon((0, 8), (3, 8), (3, 10), (0, 10)) → poly15, d_{13}, e_{13}, f_{13}, g_{13}
  50. Polygon((15, 6), (18, 6), (18, 8), (15, 8)) → poly16, h_{13}, i_{13}, j_{13}, k_{13}
  51. Polygon((12, 6), (15, 6), (15, 8), (12, 8)) → poly17, l_{13}, m_{13}, n_{13}, p_{13}
  52. Polygon((9, 6), (12, 6), (12, 8), (9, 8)) → poly18, q_{13}, r_{13}, s_{13}, t_{13}
  53. Polygon((6, 6), (9, 6), (9, 8), (6, 8)) → poly19, a_{14}, b_{14}, c_{14}, d_{14}
  54. Polygon((3, 6), (6, 6), (6, 8), (3, 8)) → poly20, e_{14}, f_{14}, g_{14}, h_{14}
  55. Polygon((0, 6), (3, 6), (3, 8), (0, 8)) → poly21, i_{14}, j_{14}, k_{14}, l_{14}
  56. Polygon((2, 11), (2.5, 10), (3.5, 10.5), (4.5, 10), (4, 11), (4.5, 12), (3.5, 12.5), (2.5, 12)) → poly46, b_{11}, c_{11}, d_{11}, e_{11}, f_{11}, g_{11}, h_{11}, i_{11}
  57. Polygon((2, 7), (2.5, 6), (3.5, 6.5), (4.5, 6), (4, 7), (4.5, 8), (3.5, 8.5), (2.5, 8)) → poly3, i, j, l, m, n, p, q, s
  58. Polygon((0, 11), (0.5, 10), (1.5, 10.5), (2.5, 10), (2, 11), (2.5, 12), (1.5, 12.5), (0.5, 12)) → poly23, t, a_1, b_1, c_1, d_1, e_1, f_1, g_1
  59. Polygon((14, 9), (14.5, 8), (15.5, 8.5), (16.5, 8), (16, 9), (16.5, 10), (15.5, 10.5), (14.5, 10)) → poly24, h_1, i_1, j_1, k_1, l_1, m_1, n_1, p_1
  60. Polygon((12, 9), (12.5, 8), (13.5, 8.5), (14.5, 8), (14, 9), (14.5, 10), (13.5, 10.5), (12.5, 10)) → poly25, q_1, r_1, s_1, t_1, a_2, b_2, c_2, d_2
  61. Polygon((10, 9), (10.5, 8), (11.5, 8.5), (12.5, 8), (12, 9), (12.5, 10), (11.5, 10.5), (10.5, 10)) → poly26, e_2, f_2, g_2, h_2, i_2, j_2, k_2, l_2
  62. Polygon((8, 9), (8.5, 8), (9.5, 8.5), (10.5, 8), (10, 9), (10.5, 10), (9.5, 10.5), (8.5, 10)) → poly27, m_2, n_2, p_2, q_2, r_2, s_2, t_2, a_3
  63. Polygon((6, 9), (6.5, 8), (7.5, 8.5), (8.5, 8), (8, 9), (8.5, 10), (7.5, 10.5), (6.5, 10)) → poly28, b_3, c_3, d_3, e_3, f_3, g_3, h_3, i_3
  64. Polygon((4, 9), (4.5, 8), (5.5, 8.5), (6.5, 8), (6, 9), (6.5, 10), (5.5, 10.5), (4.5, 10)) → poly29, j_3, k_3, l_3, m_3, n_3, p_3, q_3, r_3
  65. Polygon((2, 9), (2.5, 8), (3.5, 8.5), (4.5, 8), (4, 9), (4.5, 10), (3.5, 10.5), (2.5, 10)) → poly30, s_3, t_3, a_4, b_4, c_4, d_4, e_4, f_4
  66. Polygon((0, 9), (0.5, 8), (1.5, 8.5), (2.5, 8), (2, 9), (2.5, 10), (1.5, 10.5), (0.5, 10)) → poly31, g_4, h_4, i_4, j_4, k_4, l_4, m_4, n_4
  67. Polygon((14, 7), (14.5, 6), (15.5, 6.5), (16.5, 6), (16, 7), (16.5, 8), (15.5, 8.5), (14.5, 8)) → poly32, p_4, q_4, r_4, s_4, t_4, a_5, b_5, c_5
  68. Polygon((12, 7), (12.5, 6), (13.5, 6.5), (14.5, 6), (14, 7), (14.5, 8), (13.5, 8.5), (12.5, 8)) → poly33, d_5, e_5, f_5, g_5, h_5, i_5, j_5, k_5
  69. Polygon((10, 7), (10.5, 6), (11.5, 6.5), (12.5, 6), (12, 7), (12.5, 8), (11.5, 8.5), (10.5, 8)) → poly34, l_5, m_5, n_5, p_5, q_5, r_5, s_5, t_5
  70. Polygon((8, 7), (8.5, 6), (9.5, 6.5), (10.5, 6), (10, 7), (10.5, 8), (9.5, 8.5), (8.5, 8)) → poly35, a_6, b_6, c_6, d_6, e_6, f_6, g_6, h_6
  71. Polygon((6, 7), (6.5, 6), (7.5, 6.5), (8.5, 6), (8, 7), (8.5, 8), (7.5, 8.5), (6.5, 8)) → poly36, i_6, j_6, k_6, l_6, m_6, n_6, p_6, q_6
  72. Polygon((4, 7), (4.5, 6), (5.5, 6.5), (6.5, 6), (6, 7), (6.5, 8), (5.5, 8.5), (4.5, 8)) → poly37, r_6, s_6, t_6, a_7, b_7, c_7, d_7, e_7
  73. Polygon((0, 7), (0.5, 6), (1.5, 6.5), (2.5, 6), (2, 7), (2.5, 8), (1.5, 8.5), (0.5, 8)) → poly38, f_7, g_7, h_7, i_7, j_7, k_7, l_7, m_7
  74. Polygon((14, 11), (14.5, 10), (15.5, 10.5), (16.5, 10), (16, 11), (16.5, 12), (15.5, 12.5), (14.5, 12)) → poly39, n_7, p_7, q_7, r_7, s_7, t_7, a_8, b_8
  75. Polygon((12, 11), (12.5, 10), (13.5, 10.5), (14.5, 10), (14, 11), (14.5, 12), (13.5, 12.5), (12.5, 12)) → poly40, c_8, d_8, e_8, f_8, g_8, h_8, i_8, j_8
  76. Polygon((10, 11), (10.5, 10), (11.5, 10.5), (12.5, 10), (12, 11), (12.5, 12), (11.5, 12.5), (10.5, 12)) → poly41, k_8, l_8, m_8, n_8, p_8, q_8, r_8, s_8
  77. Polygon((8, 11), (8.5, 10), (9.5, 10.5), (10.5, 10), (10, 11), (10.5, 12), (9.5, 12.5), (8.5, 12)) → poly42, t_8, a_9, b_9, c_9, d_9, e_9, f_9, g_9
  78. Polygon((6, 11), (6.5, 10), (7.5, 10.5), (8.5, 10), (8, 11), (8.5, 12), (7.5, 12.5), (6.5, 12)) → poly43, h_9, i_9, j_9, k_9, l_9, m_9, n_9, p_9
  79. Polygon((4, 11), (4.5, 10), (5.5, 10.5), (6.5, 10), (6, 11), (6.5, 12), (5.5, 12.5), (4.5, 12)) → poly44, q_9, r_9, s_9, t_9, a_{10}, b_{10}, c_{10}, d_{10}
  80. Point(list2) → E'
  81. If(Div[13, side] ≟ 13 / side, 13 / side - 1, Div[13, side]) → max_i
  82. If(Div[5, side] ≟ 5 / side, 5 / side - 1, Div[5, side]) → max_j
  83. Sequence(Segment[E + (i * side, 0), E + (i * side, (max_j + 1) * side)], i, 0, max_i + 1) → list1
  84. Sequence(Segment[E + (0, j * side), E + ((max_i + 1) * side, j * side)], j, 0, max_j + 1) → list1'
  85. Polygon(E, (x(F), y(E)), F, (x(E), y(F))) → poly22, e_{15}, m_{14}, f_{15}, n_{14}