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IM 6.1.13 Lesson: Polyhedra

IM 6.1.13 Lesson: Polyhedra
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2020-07-14 19:35
PolygonSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates the basic structure and properties of polyhedra, including prisms, pyramids, cubes, and tetrahedra. The core mathematical concepts involve solid geometry: definitions of faces, edges, vertices, and an intuitive understanding of Euler's formula. The construction uses GeoGebra commands such as Segment, Point, Polygon, Midpoint, Cube, Tetrahedron, Pyramid, Prism, Centroid, OrthogonalLine, Circle, Intersect, Plane, and PointIn to build multiple 3D objects and auxiliary points, lines, and planes, with interactive buttons to explore different polyhedra. It is suitable for middle school mathematics (e.g., grade 6) to help students recognize polyhedra, distinguish between prisms and pyramids, and develop spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(C, D) → g
  3. Point(yAxis) → E
  4. Point(xAxis) → F
  5. Point(yAxis) → G
  6. Polygon(E, F, G, H) → q1, e, f_1, g_1, h
  7. Midpoint(f_1) → I
  8. Midpoint(h) → J
  9. Polygon(K, G, F) → t1, f_2, k, g_2
  10. Polygon(K, L, H, G) → q2, k_1, l, h_1, g_3
  11. Polygon(K, L, E, F) → q3, k_2, l_1, e_1, f_3
  12. Polygon(L, H, E) → t2, e_2, l_2, h_2
  13. Point(xAxis) → M
  14. Point(yAxis) → N
  15. Point(Circle[N, Distance[M, N], Segment[M, N]]) → O
  16. Cube(M, N, O) → a, P, faceOPTS, faceQRST, edgeMP, edgeMN, edgeNO, edgeOP, edgeMQ, edgePT, edgeQT, edgeQR, Q, edgeNR, edgeRS, edgeOS, edgeST, R, S, T, faceMNOP, faceMPTQ, faceMNRQ, faceNOSR
  17. Polygon(U, V, W, Z) → q4, u, v, w, z_1
  18. Polygon(C_1, D_1, Z, W) → q5, c_1, d_1, z_2, w_1
  19. Polygon(A_1, B_1, V, U) → q7, j, b_1, v_1, u_2
  20. Polygon(D_1, A_1, B_1, C_1) → q8, m, p, q, r
  21. Polygon(E_1, F_1, G_1, H_1) → q9, s, t, b, c
  22. Polygon(J_1, I_1, E_1, F_1) → q12, s_1, t_1, i_2, j_2
  23. Polygon(J_1, K_1, N_1, I_1) → q13, m_2, n_2, p_2, q_2
  24. Polygon(K_1, J_1, F_1, G_1) → q14, r_2, s_2, t_2, a_2
  25. Polygon(K_1, N_1, H_1, G_1) → q6, i, n_1, d, i_1
  26. Polygon(I_1, E_1, H_1, N_1) → q10, j_1, m_1, p_1, q_1
  27. Point(Circle[Midpoint[O_1, P_1], (Distance[O_1, P_1] * sqrt(3) / 2), Segment[O_1, P_1]]) → Q_1
  28. Tetrahedron(O_1, P_1, Q_1) → o, R_1, edge6, edge5, face1, face3, face2, face4, edge2, edge1, edge4, edge3
  29. Point(xAxis) → S_1
  30. Point(yAxis) → T_1
  31. Polygon(S_1, T_1, 4) → poly1, r_1, a_1, b_2, c_2, U_1, V_1
  32. Pyramid(poly1, 3.4903016730249985) → o_1, W_1, face5, face7, face8, face6, edge7, edge8, edge9, edge10
  33. Polygon(Z_1, A_2, 5, xOyPlane) → poly2, d_2, h_3, i_3, j_3, k_3, B_2, C_2, D_2
  34. Pyramid(poly2, 4) → u_1, E_2, edge14, edge15, face9, face11, face12, face13, face10, edge11, edge12, edge13
  35. Point(yAxis) → F_2
  36. Point(yAxis) → G_2
  37. Polygon(F_2, G_2, 3) → poly3, l_3, m_3, n_3, H_2
  38. Prism(poly3, 3) → o_2, I_2, edge18, edge19, edge21, edge20, J_2, K_2, face15, face17, face16, face14, edge16, edge17
  39. Polygon(J_2, N_2, G_2) → t3, p_3, q_3, r_3
  40. Polygon(I_2, M_2, F_2) → t4, s_3, t_3, a_3
  41. Polygon(M_2, N_2, J_2, I_2) → q11, b_3, c_3, d_3, e_3
  42. Polygon(M_2, N_2, G_2, F_2) → q15, f_4, g_4, h_4, i_4
  43. Polygon(O_2, P_2, 6, xOyPlane) → poly4, j_4, T_2, k_4, l_4, m_4, n_4, p_4, Q_2, R_2, S_2
  44. Centroid(poly4) → V_2
  45. OrthogonalLine(V_2, xOyPlane) → q_4
  46. Point(q_4) → U_2
  47. Circle(U_2, (2 * sqrt(3) / 3), xOyPlane) → c_4
  48. Segment(U_2, W_2) → r_4
  49. Intersect(r_4, c_4) → Z_2,
  50. Segment(U_2, A_3) → s_4
  51. Intersect(s_4, c_4) → B_3,
  52. Segment(U_2, C_3) → t_4
  53. Intersect(t_4, c_4) → D_3,
  54. Polygon(B_3, Q_2, R_2) → t5, a_4, b_4, d_4
  55. Polygon(D_3, S_2, T_2) → t6, e_4, f_5, g_5
  56. Polygon(Z_2, O_2, P_2) → t7, h_5, i_5, j_5
  57. Polygon(B_3, Z_2, P_2, Q_2) → q16, k_5, l_5, m_5, n_5
  58. Polygon(D_3, Z_2, O_2, T_2) → q17, p_5, q_5, r_5, s_5
  59. Polygon(B_3, D_3, S_2, R_2) → q18, t_5, a_5, b_5, c_5
  60. Polygon(Z_2, B_3, D_3) → t8, d_5, e_5, f_6
  61. Polygon(T_2, D_3, S_2) → t9, g_6, h_6, i_6
  62. Polygon(Q_2, B_3, R_2) → t10, j_6, k_6, l_6
  63. Polygon(Z, Z_2, O_2) → t11, m_6, z_3, n_6
  64. Segment(E_3, F_3) → p_6
  65. Point(yAxis) → H_3
  66. Polygon(G_3, H_3, 5, xOyPlane) → poly5, q_6, r_6, s_6, t_6, a_6, I_3, J_3, K_3
  67. Centroid(poly5) → M_3
  68. Plane(L_3, poly5) → p_7
  69. Polygon(O_3, N_3, 5, xOyPlane) → poly6, b_6, c_6, d_6, e_6, f_7, P_3, Q_3, R_3
  70. Polygon(P_3, H_3, I_3) → t12, g_7, h_7, i_7
  71. Polygon(P_3, N_3, H_3) → t13, j_7, k_7, l_7
  72. Polygon(P_3, Q_3, I_3) → t14, m_7, n_7, q_7
  73. Polygon(Q_3, I_3, J_3) → t15, r_7, s_7, t_7
  74. Polygon(Q_3, R_3, J_3) → t16, a_7, b_7, c_7
  75. Polygon(R_3, K_3, J_3) → t17, d_7, e_7, f_8
  76. Polygon(R_3, K_3, O_3) → t18, o_3, g_8, h_8
  77. Polygon(K_3, O_3, G_3) → t19, i_8, j_8, k_8
  78. Polygon(G_3, N_3, O_3) → t20, l_8, m_8, n_8
  79. Polygon(G_3, H_3, N_3) → t21, p_8, q_8, r_8
  80. Intersect(q, b_1) → S_3
  81. Intersect(q, w_1) → T_3
  82. Point(w_1) → U_3
  83. Intersect(b_1, v) → V_3
  84. Polygon(S_3, T_3, U_3, V_3) → q19, s_8, t_8, u_3, v_3
  85. Intersect(m, d_1) → W_3
  86. Intersect(m, u_2) → Z_3
  87. Intersect(u_2, v_1) → A_4
  88. Intersect(z_1, d_1) → B_4
  89. Polygon(W_3, Z_3, A_4, B_4) → q20, w_3, a_8, b_8, c_8
  90. Point(q_8) → C_4
  91. PointIn(poly5) → D_4
  92. Polygon(C_4, I, D_4) → t22, d_8, e_8, i_9
  93. PointIn(poly5) → E_4
  94. Polygon(I, E_4, C_4) → t23, f_9, i_{10}, g_9

AI Drawing Prompt Reference

Draw a cube with 6 faces, 12 edges, and 8 vertices. Then draw a tetrahedron with 4 faces and 4 vertices. Finally, draw a triangular prism with 2 bases and 3 sides.
The prompt above was compiled from the original drawing commands and is for reference only.