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Exploring Volume by Janet Bowers

Exploring Volume by Janet Bowers
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2014-09-12 03:28
Solid GeometryCalculusGeometric Construction
This example demonstrates the concept of volume through geometric construction. It involves core concepts from integral calculus and solid geometry. Using GeoGebra commands such as Circle, Vector, Translate, Intersect, etc., users can dynamically explore the volumes of different shapes. Suitable for high school or university calculus and solid geometry courses.

GeoGebra Commands & Parameters

  1. Circle(Origin, len - (2 * h)) → c
  2. Vector((width - (2 * h), 0)) → u
  3. Translate(A, u) → A'
  4. Intersect(c, xAxis) → B, C
  5. Rotate(C, α) → C'
  6. Vector(C') → v
  7. Translate(A, v) → A'_1
  8. Translate(A', v) → A''
  9. Polygon(A, A'_1, A'', A') → poly1, a, a'_1, a'', a'
  10. Vector((0, h)) → w
  11. Translate(A, w) → A'_2
  12. Translate(A', w) → A''_1
  13. Translate(A'_1, w) → D
  14. Translate(A'', w) → A'''
  15. Segment(A'_2, A) → b
  16. Segment(A''_1, A') → d
  17. Segment(D, A'_1) → e
  18. Segment(A''', A'') → f
  19. Polygon(A'_2, A''_1, A', A) → poly2, a'_2, a''_1, a'_3, a_1
  20. Polygon(D, A''', A'', A'_1) → poly3, d_1, a''', a''_2, a'_4
  21. Polygon(A''_1, A''', A'', A') → poly4, a''_3, a'''_1, a''_4, a'_5
  22. Polygon(D, A'_2, A, A'_1) → poly5, d_2, a'_6, a_2, a'_7
  23. Rotate(E, α) → E'
  24. Vector(E') → g
  25. Segment(W_1, A_2) → c_7
  26. Segment(A_2, Z_1) → d_7
  27. Segment(Z_1, V_1) → e_7
  28. Segment(V_1, W_1) → f_7
  29. Segment(Paper, U_1) → g_7
  30. Segment(U_1, S_1) → h_7
  31. Segment(S_1, T_1) → i_7
  32. Segment(T_1, Paper) → j_7
  33. OrthogonalLine(Z_1, g_7) → m_7
  34. OrthogonalLine(Z_1, h_7) → n_7
  35. OrthogonalLine(A_2, h_7) → a_8
  36. OrthogonalLine(W_1, i_7) → b_8
  37. Intersect(j_7, a_8) → B_2
  38. Intersect(i_7, b_8) → C_2
  39. Intersect(m_7, i_7) → D_2
  40. Intersect(h_7, a_8) → E_2
  41. Intersect(n_7, h_7) → F_2
  42. Intersect(m_7, g_7) → G_2
  43. Intersect(g_7, b_8) → H_2
  44. Intersect(n_7, j_7) → I_2
  45. Polygon(B_2, W_1, 4) → poly26, c_8, d_8, e_8, f_8, J_2, K_2
  46. Polygon(Paper, H_2, 4) → poly27, g_8, h_8, i_8, j_8, L_2, M_2
  47. Polygon(G_2, U_1, 4) → poly28, k_8, l_8, m_8, n_8, N_2, O_2
  48. Polygon(A_2, E_2, 4) → poly29, p_8, q_8, r_8, s_8, P_2, Q_2
  49. Polygon(V_1, Z_1, A_2, W_1) → poly31, v_1, z_1, a_9, w_1
  50. newbox(α, E, G) → n, p, c_1, e_1, f_1, g_1, h_1, i_1, j_1, k_1, J, q, H, poly6, poly7, r, s, t, b_1
  51. Intersect(r, i_1) → K
  52. Intersect(r, i_1) → L
  53. Intersect(r, e_1) → M
  54. Polygon(J, M, H, G) → poly8, j_2, m_1, h_2, g_2
  55. Polygon(B_2, W_1, V_1, I_2) → poly9, b_2, w_2, v_2, i_2
  56. Polygon(W_1, A_2, D_2, C_2) → poly10, w_3, a_3, d_3, c_2
  57. Point(w_3) → N
  58. Polygon(N, Z_1, F_2, E_2) → poly11, n_1, z_2, f_2, e_2
  59. Point(n_1) → O
  60. Point(w_2) → P
  61. Polygon(H_2, G_2, O, P) → poly12, h_3, g_3, o, p_1
  62. blackbox(α, E, Q) → poly13, poly14, r_1, s_1, t_1, k_2, l_2, m_2, n_2, p_2, q_2, r_2, poly15, s_2, t_2, i, j, k, l, m, l_1, q_1
  63. blackbox(α, E, R) → poly16, poly17, l_3, m_3, n_3, p_3, q_3, r_3, s_3, t_3, a_4, b_4, poly18, c_4, d_4, b_3, c_3, e_3, f_3, i_3, j_3, k_3
  64. blackbox(α, E, S) → poly19, poly20, l_4, m_4, n_4, p_4, q_4, r_4, s_4, t_4, a_5, b_5, poly21, c_5, d_5, e_4, f_4, g_4, h_4, i_4, j_4, k_4
  65. blackbox(α, E, T) → poly22, poly23, l_5, m_5, n_5, p_5, q_5, r_5, s_5, t_5, a_6, b_6, poly24, c_6, d_6, e_5, f_5, g_5, h_5, i_5, j_5, k_5
  66. blackbox(α, E, V) → poly33, poly34, c_9, d_9, e_9, f_9, g_9, h_9, i_9, j_9, k_9, l_9, poly35, m_9, n_9, p_7, q_7, r_7, s_7, t_7, t_8, b_9
  67. blackbox(α, E, W) → poly36, poly37, c_{10}, d_{10}, e_{10}, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}, poly38, m_{10}, n_{10}, p_9, q_9, r_9, s_9, t_9, a_{10}, b_{10}
  68. blackbox(α, E, A_1) → poly42, poly43, c_{12}, d_{12}, e_{12}, f_{12}, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, l_{12}, poly44, m_{12}, n_{12}, p_{11}, q_{11}, r_{11}, s_{11}, t_{11}, a_{12}, b_{12}
  69. blackbox(α, E, B_1) → poly45, poly46, c_{13}, d_{13}, e_{13}, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, k_{13}, l_{13}, poly47, m_{13}, n_{13}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}, a_{13}, b_{13}
  70. blackbox(α, E, C_1) → poly48, poly49, c_{14}, d_{14}, e_{14}, f_{14}, g_{14}, h_{14}, i_{14}, j_{14}, k_{14}, l_{14}, poly50, m_{14}, n_{14}, p_{13}, q_{13}, r_{13}, s_{13}, t_{13}, a_{14}, b_{14}
  71. blackbox(α, E, D_1) → poly51, poly52, c_{15}, d_{15}, e_{15}, f_{15}, g_{15}, h_{15}, i_{15}, j_{15}, k_{15}, l_{15}, poly53, m_{15}, n_{15}, p_{14}, q_{14}, r_{14}, s_{14}, t_{14}, a_{15}, b_{15}
  72. blackbox(α, E, E_1) → poly54, poly55, c_{16}, d_{16}, e_{16}, f_{16}, g_{16}, h_{16}, i_{16}, j_{16}, k_{16}, l_{16}, poly56, m_{16}, n_{16}, p_{15}, q_{15}, r_{15}, s_{15}, t_{15}, a_{16}, b_{16}
  73. blackbox(α, E, F_1) → poly57, poly58, c_{17}, d_{17}, e_{17}, f_{17}, g_{17}, h_{17}, i_{17}, j_{17}, k_{17}, l_{17}, poly59, m_{17}, n_{17}, p_{16}, q_{16}, r_{16}, s_{16}, t_{16}, a_{17}, b_{17}
  74. blackbox(α, E, G_1) → poly60, poly61, c_{18}, d_{18}, e_{18}, f_{18}, g_{18}, h_{18}, i_{18}, j_{18}, k_{18}, l_{18}, poly62, m_{18}, n_{18}, p_{17}, q_{17}, r_{17}, s_{17}, t_{17}, a_{18}, b_{18}
  75. Segment(A''_1, A''') → e_6
  76. Segment(A'_2, A''_1) → f_6
  77. Polygon(T_2, V_2, W_2, U_2) → poly25, t_6, v_6, w_5, u_6
  78. Polygon(S_2, Z_2, B_3, A_3) → poly30, s_6, z_3, b_7, a_7

AI Drawing Prompt Reference

Draw a solid geometry figure, starting with a circle c, a width vector u passing through the center of the circle, and a rectangle formed by the intersection points B and C of the circle with the x-axis. Use rotations and translations to create multiple parallelograms and trapezoids, ultimately forming a complex 3D shape with multiple faces and edges.
The prompt above was compiled from the original drawing commands and is for reference only.