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Geometric Representation of (a + b) Cubed

Geometric Representation of (a + b) Cubed
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2020-01-03 16:35
Solid GeometryGeometric Construction
This example demonstrates the geometric representation of (a + b) cubed, illustrating volume calculations and solid geometry concepts. It involves constructing prisms, spheres, and polygons using GeoGebra commands like Point, Cube, Line, Intersect, and more. Ideal for high school students learning polynomial volume formulas and geometric relationships.

GeoGebra Commands & Parameters

  1. Point(Circle[J, Distance[L, J], Segment[L, J]]) → K
  2. Cube(L, J, K) → i, M, faceKMQP, faceNOPQ, edgeLM, edgeJL, edgeJK, edgeKM, edgeLN, edgeMQ, edgeNQ, edgeNO, N, edgeJO, edgeOP, edgeKP, edgePQ, O, P, Q, faceJKML, faceLMQN, faceJLNO, faceJKPO
  3. Line(A, xAxis) → h
  4. Line(A, zAxis) → f
  5. Intersect(f, faceNOPQ) → R
  6. Line(R, h) → j
  7. Intersect(edgeOP, j) → U
  8. Sphere(A, d) → e
  9. Line(A, yAxis) → g
  10. Intersect(f, e) → B, C
  11. Intersect(h, e) → D, E
  12. Intersect(g, e) → F, G
  13. Line(R, g) → k
  14. Intersect(edgeNO, k) → S
  15. Distance(R, U) → XR
  16. Intersect(edgeNQ, j) → T
  17. Distance(R, A) → ZR
  18. Distance(T, Q) → YR
  19. Line(V, j) → l
  20. Line(G, l) → m
  21. Line(E, g) → n
  22. Line(W, g) → p
  23. Intersect(m, n) → Z
  24. Intersect(l, n) → A_1
  25. Intersect(l, p) → B_1
  26. Intersect(m, p) → C_1
  27. Polygon(Z, C_1, B_1, A_1) → q1, z_1, c_1, b_1, a_1
  28. Line(D, n) → q
  29. Intersect(m, q) → F_1
  30. Intersect(l, q) → G_1
  31. Line(D_1, q) → r
  32. Intersect(r, l) → E_1
  33. Intersect(m, r) → H_1
  34. Polygon(E_1, H_1, F_1, G_1) → q2, e_1, h_1, f_1, g_1
  35. Line(F, h) → s
  36. Line(I_1, s) → t
  37. Intersect(r, s) → J_1
  38. Intersect(q, s) → K_1
  39. Intersect(q, t) → L_1
  40. Intersect(r, t) → M_1
  41. Intersect(k, s) → N_1
  42. Intersect(t, p) → O_1
  43. Intersect(s, p) → P_1
  44. Intersect(s, n) → Q_1
  45. Intersect(t, n) → R_1
  46. Polygon(M_1, J_1, K_1, L_1) → q3, m_1, j_1, k_1, l_1
  47. Polygon(R_1, Q_1, P_1, O_1) → q4, r_1, q_1, p_1, o_1
  48. OrthogonalLine(A_1, q1) → i_1
  49. OrthogonalLine(B_1, q1) → n_1
  50. OrthogonalLine(C_1, q1) → s_1
  51. OrthogonalLine(Z, q1) → t_1
  52. OrthogonalLine(E_1, q2) → d_1
  53. OrthogonalLine(G_1, q2) → f_2
  54. OrthogonalLine(H_1, q2) → g_2
  55. OrthogonalLine(F_1, q2) → h_2
  56. OrthogonalPlane(C, f) → p_2
  57. Intersect(d_1, p_2) → S_1
  58. Intersect(g_2, p_2) → T_1
  59. Intersect(h_2, p_2) → U_1
  60. Intersect(f_2, p_2) → V_1
  61. Intersect(t_1, p_2) → W_1
  62. Intersect(i_1, p_2) → Z_1
  63. Intersect(s_1, p_2) → A_2
  64. Intersect(n_1, p_2) → B_2
  65. Polygon(W_1, A_2, B_2, Z_1) → q5, w_1, a_2, b_2, i_2
  66. Polygon(T_1, U_1, V_1, S_1) → q6, j_2, u_1, v_1, k_2
  67. OrthogonalLine(O_1, q4) → l_2
  68. OrthogonalLine(P_1, q4) → m_2
  69. OrthogonalLine(Q_1, q4) → n_2
  70. OrthogonalLine(R_1, q4) → q_2
  71. OrthogonalLine(J_1, q3) → r_2
  72. OrthogonalLine(M_1, q3) → s_2
  73. OrthogonalLine(L_1, q3) → t_2
  74. OrthogonalLine(K_1, q3) → c_2
  75. Intersect(l_2, p_2) → C_2
  76. Intersect(m_2, p_2) → D_2
  77. Intersect(n_2, p_2) → E_2
  78. Intersect(q_2, p_2) → F_2
  79. Intersect(s_2, p_2) → G_2
  80. Intersect(t_2, p_2) → H_2
  81. Intersect(r_2, p_2) → I_2
  82. Intersect(c_2, p_2) → J_2
  83. Polygon(G_2, H_2, J_2, I_2) → q7, d_2, e_2, f_3, g_3
  84. Polygon(F_2, C_2, D_2, E_2) → q8, h_3, i_3, j_3, k_3
  85. OrthogonalPlane(B, f) → p_3
  86. Intersect(n_1, p_3) → K_2
  87. Intersect(i_1, p_3) → L_2
  88. Intersect(t_1, p_3) → M_2
  89. Intersect(s_1, p_3) → N_2
  90. Intersect(f_2, p_3) → O_2
  91. Intersect(d_1, p_3) → P_2
  92. Intersect(g_2, p_3) → Q_2
  93. Intersect(h_2, p_3) → R_2
  94. Polygon(K_2, L_2, M_2, N_2) → q9, l_3, m_3, n_3, q_3
  95. Polygon(O_2, P_2, Q_2, R_2) → q10, o_2, r_3, s_3, t_3
  96. Intersect(r_2, p_3) → S_2
  97. Intersect(c_2, p_3) → T_2
  98. Intersect(t_2, p_3) → U_2
  99. Intersect(s_2, p_3) → V_2
  100. Intersect(n_2, p_3) → W_2
  101. Intersect(q_2, p_3) → Z_2
  102. Intersect(m_2, p_3) → A_3
  103. Intersect(l_2, p_3) → B_3
  104. Polygon(Z_2, W_2, A_3, B_3) → q11, z_2, w_2, a_3, b_3
  105. Polygon(U_2, T_2, S_2, V_2) → q12, u_2, c_3, d_3, v_2
  106. Prism(q6, ZR) → v, G_3, edge10, edge11, edge12, edge9, edge13, edge15, edge16, edge14, H_3, I_3, J_3, face8, face9, face7, face6, face10
  107. Prism(q7, ZR) → w, K_3, edge17, edge18, edge20, edge19, edge21, edge23, edge24, edge22, L_3, M_3, N_3, face11, face13, face14, face12, face15
  108. Prism(q8, ZR) → e_3, O_3, edge28, edge25, edge26, edge27, edge29, edge31, edge32, edge30, P_3, Q_3, R_3, face17, face16, face18, face19, face20
  109. Prism(q5, ZR) → o_3, S_3, edge33, edge35, edge36, edge34, edge37, edge39, edge40, edge38, T_3, U_3, V_3, face22, face24, face23, face21, face25
  110. Prism(q9, u_3) → v_3, W_3, edge41, edge42, edge43, edge44, edge45, edge47, edge48, edge46, Z_3, A_4, B_4, face26, face28, face29, face27, face30
  111. Prism(q11, ZL) → w_3, C_4, edge50, edge49, edge51, edge52, edge53, edge55, edge56, edge54, D_4, E_4, F_4, face31, face32, face34, face33, face35
  112. Prism(q10, a_4) → b_4, G_4, edge57, edge58, edge59, edge60, edge61, edge63, edge64, edge62, H_4, I_4, J_4, face36, face38, face39, face37, face40
  113. Prism(q12, c_4) → d_4, K_4, edge67, edge66, edge65, edge68, edge69, edge71, edge72, edge70, L_4, M_4, N_4, face43, face41, face42, face44, face45
  114. Midpoint(edge58) → D_3
  115. Midpoint(edge9) → C_9
  116. Midpoint(edge14) → C_10
  117. Midpoint(edge22) → E10
  118. Midpoint(edge21) → E_3
  119. Midpoint(T_3, U_3) → C_3

AI Drawing Prompt Reference

Draw a cube and a sphere inside it, with the sphere intersecting one face of the cube at two points. Add the diagonals of the cube. Then, on one face of the cube, draw two line segments that intersect the diagonal at a point, and connect this point to a vertex of the cube to form two triangles. Finally, draw three triangles outside the cube, intersecting its three faces and connecting the intersection points.
The prompt above was compiled from the original drawing commands and is for reference only.