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Can You Find the Volume?

Can You Find the Volume?
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2018-04-17 19:29
Solid GeometryCalculusGeometric Construction
This example demonstrates how to construct prisms and pyramids using GeoGebra and calculate their volumes. Key math concepts include volume calculation, cross-sectional methods, polygon construction, and relationships between segments and points. The construction involves commands and objects like Prism, Pyramid, Intersect, Point, Midpoint, Segment, Polygon, AngularBisector, PointIn, and Distance. It is suitable for high school solid geometry instruction, helping students understand volume calculation methods and spatial geometric relationships.

GeoGebra Commands & Parameters

  1. Prism(Polygon[(((-b_1)) / 2, ((-b_1)) / 2), (b_1 / 2, ((-b_1)) / 2), 4], (((-b_1)) / 2, ((-b_1)) / 2, b_1)) → a, A, edge2, edge3, edge4, edge5, edge6, edge7, edge8, B, C, face1_1, face2_1, face3_1, face4, face5, edge1
  2. Prism(Polygon[(-3, -5, 0), (-3, 5, 0), (-3, -5 + s_4, 4.8)], (3, -5, 0)) → d, H, edge28, edge29, edge30, I, face18, face19, face20, face21, edge25, edge26, edge27
  3. Prism(Polygon[(((-b_2)) / 2, ((-b_2)) / 2), (b_2 / 2, ((-b_2)) / 2), 5], (((-b_2)) / 2, ((-b_2)) / 2, h_2)) → b, D, face11, edge9, edge10, edge11, edge12, edge13, edge14, edge15, edge16, edge17, E, edge18, F, G, face6, face7, face8, face9, face10
  4. Pyramid(Polygon[((-b_3), 0), Rotate[((-b_3), 0), 60°, (0, 0)], 6], (0, 0, h_3)) → c, face12, edge22, edge23, edge24, face14, face15, face16, face17, face13, edge19, edge20, edge21
  5. Prism(Polygon[(((-b_5)) / 2, (-b'_5)), (((-b_5)) / 2, 0), (0, 18 / b_5), (b_5 / 2, 0), (b_5 / 2, (-b'_5))], (((-b_5)) / 2, (-b'_5), s_5)) → e, J, face27, edge31, edge32, edge33, edge34, edge35, edge36, edgeJK, edgeKL, edgeLM, K, edge37, L, M, face22, face24, face25, face26, face23
  6. Prism(Polygon[(0, 2.5), (7, 0), (0, -2.5), (-7, 0)], (0, 2.5, s_6)) → f, N, edge39, edge40, edge41, edge42, edgeNO, edgeOP, edge43, O, P, face28, face30, face31, face29, face32, edge38
  7. Intersect(xAxis, yAxis) → Q
  8. Point(xAxis) → R
  9. Point(xAxis) → S
  10. Midpoint(edge11) → Z
  11. Point(edge12) → A_1
  12. Segment(E, A_1) → g
  13. Midpoint(g) → B_1
  14. Intersect(edge6, edge2) → C_1
  15. Point(edge3) → D_1
  16. Segment(C_1, D_1) → h
  17. Midpoint(h) → G_1
  18. Polygon(A_2, B_2, C_2) → poly1_1, c_1, a_2, b_4
  19. Polygon(D_2, E_2, F_2) → poly2_1, f_1, d_1, e_1
  20. Polygon(A_2, E_2, H_1) → poly3_1, h_1, a_3, e_2
  21. Polygon(B_2, F_2, G_2) → poly4_1, g_1, b_6, f_2
  22. Polygon(B_2, C_2, D_2, F_2) → poly5_1, b_7, c_2, d_2, f_3
  23. Polygon(A_2, C_2, D_2, E_2) → poly6_1, a_4, c_3, d_3, e_3
  24. Polygon(A_2, B_2, G_2, H_1) → poly7_1, a_5, b_8, g_2, h_4
  25. Polygon(E_2, F_2, G_2, H_1) → poly8_1, e_4, f_4, g_3, h_5
  26. Segment(G_2, A_2) → i_1
  27. Midpoint(i_1) → I_1
  28. AngularBisector(F_2, B_2, G_2) → j_1
  29. AngularBisector(G_2, F_2, B_2) → k_1
  30. Intersect(k_1, j_1) → J_1
  31. AngularBisector(A_2, B_2, D_2) → l_1
  32. AngularBisector(D_2, A_2, B_2) → m_1
  33. PointIn(face19) → Pointface19
  34. PointIn(face32) → Pointface32
  35. Intersect(edge43, edge38) → K_1
  36. Segment(K_1, O) → i
  37. Midpoint(i) → L_1
  38. Point(edge40) → M_1
  39. Segment(P, M_1) → j
  40. Midpoint(j) → N_1
  41. Point(edge18) → O_1
  42. Segment(O_1, E) → k
  43. Segment(D, G) → l
  44. Intersect(l, k) → P_1
  45. Segment(F, P_1) → m
  46. Segment(E, G) → p
  47. Segment(F, O_1) → q
  48. Segment(D, F) → r
  49. Intersect(r, p) → Q_1
  50. Segment(O_1, Q_1) → s
  51. Intersect(s, m) → R_1
  52. Segment(R_1, D) → t
  53. AngularBisector(edge29, edge30) → n_1, p_1
  54. AngularBisector(edge29, edge28) → q_1, r_1
  55. Intersect(p_1, r_1) → S_1
  56. Point(edge26) → T_1
  57. Segment(I, T_1) → s_1
  58. Midpoint(s_1) → U_1
  59. Point(list1) → V_1
  60. Segment(H, V_1) → t_1
  61. Midpoint(t_1) → W_1
  62. Point(edge25) → Z_1
  63. Segment(Z_1, I) → i_2
  64. Midpoint(i_2) → H_2
  65. Point(edge22) → J_2
  66. Segment(I_2, J_2) → j_2
  67. Midpoint(j_2) → K_2
  68. AngularBisector(edge22, edge23) → k_2, l_2
  69. Point(k_2) → L_2
  70. Intersect(l_2, k_2) → M_2
  71. Midpoint(L_2, M_2) → N_2
  72. Distance(edge21, edge22) → o
  73. Segment(K_2, M_2) → m_2
  74. Polygon((((-b_1)) / 2, ((-b_1)) / 2), (b_1 / 2, ((-b_1)) / 2), 4) → face1
  75. Polygon((((-b_2)) / 2, ((-b_2)) / 2), (b_2 / 2, ((-b_2)) / 2), 5) → face2
  76. Polygon(((-b_3), 0), Rotate[((-b_3), 0), 60°, (0, 0)], 6) → face3
  77. Polygon((-3, -5, 0), (-3, 5, 0), (-3, -5 + s_4, 4.8)) → base4, n_2, p_2, q_2
  78. Polygon((0, 2.5), (7, 0), (0, -2.5), (-7, 0)) → base5, r_2, s_2, t_2, i_3