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Pyramid Truncated Demonstration

Pyramid Truncated Demonstration
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2020-05-02 00:29
Coordinate GeometrySolid GeometryProbability & StatisticsInequalityGeometric Construction
This example demonstrates the geometric construction and properties of a truncated pyramid, involving core mathematical concepts such as volume calculation, centroid determination, analytical geometry, conditional probability, and inequalities. It utilizes GeoGebra commands and objects like Intersect, Polygon, Pyramid, Centroid, Segment, Translate, and Midpoint. Suitable for high school and advanced mathematics education, covering solid geometry, coordinate geometry, and probability statistics.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → C
  2. Polygon(C, C', N, xOyPlane) → poly1, j, D, F, I, J, K, k, m, n, f, i, p, q, B
  3. Pyramid(poly1, h_1) → a, A_2, edgeAC, edgeAC', edgeAB, edgeAD, edge1, edge2, edge3, edge4, faceACC', faceABC', faceABD, faceACD, face1, face2, face3, face4
  4. Centroid(poly1) → E
  5. Segment(C, C') → l_1
  6. IntersectPath(r, faceABC') → g
  7. Intersect(r, faceABC') → G, L
  8. IntersectPath(r, faceACC') → h
  9. Intersect(r, faceACC') → M, O
  10. IntersectPath(r, a) → poly2, P, b, c, d, m_1, f_1, g_1, i_1, Q, R, S, T, U, V, W, s
  11. Pyramid(poly2, A_2) → e, faceAPQ, edgeAQ, edgeAR, edgeAS, edge5, edge6, edge7, edge8, faceAQR, faceARS, faceAPS, face5, face6, face7, face8, edgeAP
  12. Segment(E, Z) → j_1
  13. Segment(S, D) → n_1
  14. Segment(A, C) → r_1
  15. Segment(A_2, E) → q_1
  16. Segment(A_2, Z) → s_1
  17. Translate(e, Vector[(t * u)]) → e'
  18. Translate(A_2, Vector[(t * u)]) → A'
  19. Translate(A_2, Vector[(t * u)]) → A'_1
  20. Translate(Z, Vector[(t * u)]) → Z'
  21. Segment(A'_1, Z') → t_1
  22. Translate(Z, Vector[(t * u)]) → Z'_1
  23. Translate(S, Vector[(t * u)]) → S'
  24. Segment(A_2, S) → b_1
  25. Translate(A_2, Vector[(t * u)]) → A'_2
  26. Translate(S, Vector[(t * u)]) → S'_1
  27. Segment(A'_2, S'_1) → c_1
  28. Segment(A_2, M) → d_1
  29. Polygon(C', B, L, G) → q1, c', b_2, l_2, g_2
  30. Polygon(B, D, S, L) → q2, b_3, d_2, s_2, l_3
  31. Polygon(D, S, T, F) → q3, d_3, s_3, t_2, f_2
  32. Polygon(T, U, I, F) → q4, t_3, u_1, i_2, f_3
  33. Polygon(I, C, M, U) → q5, i_3, c_2, m_2, u_2
  34. Polygon(C, C', G, M) → q6, c_3, h_2, g_3, m_3
  35. Polygon(B, D, S, L) → q7, b_4, d_4, s_4, l_4
  36. Polygon(B, C', G, L) → q8, b_5, n_2, g_4, l_5
  37. Polygon(C', C, M, G) → q9, p_2, c_4, m_4, g_5
  38. Polygon(M, C, J, V) → q10, m_5, c_5, j_3, v_1
  39. Polygon(J, I, U, V) → q11, j_4, i_4, u_3, v_2
  40. Polygon(U, T, F, I) → q12, u_4, t_4, f_4, i_5
  41. Polygon(F, D, S, T) → q13, f_5, d_5, s_5, t_5
  42. Polygon(B, D, S, L) → q14, b_6, d_6, s_6, l_6
  43. Polygon(B, C', G, L) → q15, b_7, a_2, g_6, l_7
  44. Polygon(C', C, M, G) → q16, e_2, c_6, m_6, g_7
  45. Polygon(C, K, W, M) → q17, c_7, k_3, w_1, m_7
  46. Polygon(D, F, T, S) → q18, d_7, f_6, t_6, s_7
  47. Polygon(F, I, U, T) → q19, f_7, i_6, u_5, t_7
  48. Polygon(I, J, V, U) → q20, i_7, j_5, v_3, u_6
  49. Polygon(K, J, V, W) → q21, k_4, j_6, v_4, w_2
  50. Polygon(C', B, L, G) → q22, p_3, b_8, l_8, g_8
  51. Polygon(B, D, S, L) → q23, b_9, d_8, s_8, l_9
  52. Polygon(D, F, T, S) → q24, d_9, f_8, t_8, s_9
  53. Polygon(C', C, M, G) → q25, q_3, c_8, m_8, g_9
  54. Polygon(L, G, C', B) → q28, l_{10}, g_{10}, e_3, b_{10}
  55. Polygon(C, C', G, M) → q29, c_{11}, h_4, g_{11}, m_{11}
  56. Polygon(D, B, L, S) → q30, d_{11}, b_{11}, l_{11}, s_{11}
  57. Segment(G, C') → p_4
  58. Segment(E, C) → a_3
  59. Segment(S, D) → a_4
  60. Segment(Z, E) → e_4
  61. Segment(D, S) → h_5
  62. Polygon(L, M, C, B) → q31, l_{12}, m_{12}, c_{12}, b_{12}
  63. Polygon(C', B, L, G) → q32, p_5, b_{13}, l_{13}, g_{12}
  64. Polygon(C, C', G, M) → q33, c_{13}, q_5, g_{13}, m_{13}
  65. Translate(poly2, Vector[(t * u)]) → poly2'
  66. Translate(T, Vector[(t * u)]) → T'
  67. Translate(U, Vector[(t * u)]) → U'
  68. Translate(V, Vector[(t * u)]) → V'
  69. Translate(W, Vector[(t * u)]) → W'
  70. Translate(M, Vector[(t * u)]) → M'
  71. Translate(G, Vector[(t * u)]) → G'
  72. Translate(L, Vector[(t * u)]) → L'
  73. Polygon(M, W, A) → t1, a_5, m_{14}, w_3
  74. Segment(Z, E) → r_5
  75. Segment(Z, E) → e_5
  76. Midpoint(M, G) → A_1
  77. Midpoint(C, C') → B_1
  78. Segment(A_1, B_1) → h_6
  79. Segment(S, D) → k_6
  80. Segment(A, B_1) → k_1
  81. Segment(A, A_1) → n_5
  82. Translate(A, Vector[(t * u)]) → A'_{3}
  83. Translate(A_1, Vector[(t * u)]) → C_1
  84. Segment(A'_{3}, C_1) → n_6
  85. Translate(Q, Vector[(t * u)]) → Q'
  86. Translate(A_2, Vector[(t * u)]) → D_1
  87. Translate(E, Vector[(t * u)]) → E'
  88. Translate(C', Vector[(t * u)]) → C''
  89. Translate(Z, Vector[(t * u)]) → Z'_{2}
  90. Translate(G, Vector[(t * u)]) → G'_{1}
  91. Translate(M, Vector[(t * u)]) → M'_{1}
  92. Translate(Z, Vector[(t * u)]) → Z'_{3}
  93. Segment(M'_{1}, Z'_{3}) → a_1
  94. Segment(A'_{3}, M'_{1}) → p_6
  95. Segment(M, Z) → r_3
  96. Segment(C, E) → n_3
  97. Point(m_2) → E_1
  98. Segment(A, E_1) → p_1
  99. Point(t_3) → F_1
  100. Segment(A, F_1) → e_1
  101. Segment(A, U) → j_2
  102. Segment(A, V) → k_2

AI Drawing Prompt Reference

Draw a truncated pyramid, showing its geometric construction and properties. First, create the base and top faces, then connect the vertices to form the lateral faces. Mark the center points of the base and top faces and draw the line connecting them. Next, select a random point on the truncated pyramid and draw lines from this point to the top and base faces, showing their intersections. Finally, translate the truncated pyramid along the line connecting the center points to display the translated version.
The prompt above was compiled from the original drawing commands and is for reference only.