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你能找到体积吗？

你能找到体积吗？
几何画板及绘图内容加载中，请稍等……
2018-04-17 19:29
立体几何微积分几何作图
①本范例演示如何通过 GeoGebra 构造棱柱、棱锥并计算其体积；②涉及的核心数学概念包括体积计算、截面法、多边形构造、线段与点的关系；③作图要点提炼自构造摘要，使用了 Prism（棱柱）、Pyramid（棱锥）、Intersect（相交）、Point（点）、Midpoint（中点）、Segment（线段）、Polygon（多边形）、AngularBisector（角平分线）、PointIn（点在对象上）、Distance（距离）等命令与对象；④适合高中立体几何教学，用于理解体积计算方法与空间几何关系。

GeoGebra 命令及参数

  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