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棱锥体积猜想的动态演示（复制）

棱锥体积猜想的动态演示（复制）
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2017-03-13 23:58
多边形立体几何积分变换滑块动画
① 本范例通过GeoGebra动态构造一个棱锥及其相关几何元素，直观演示棱锥体积与底面积、高之间的关系，并引导用户进行体积公式的猜想。② 核心数学概念包括棱锥体积公式（V = 1/3 × 底面积 × 高）、定积分在体积计算中的应用、算术平均与三角形/球体等几何体的关联，以及空间几何中的平移、旋转等变换。③ 作图要点：从构造摘要可见，使用了Sequence、Element、Point、Polygon、Segment、PointIn、Translate等命令，创建了多个点、多边形、线段和辅助对象（如z、w、v、a、O、P、α、β、sinα、cosα等），并利用滑块（echelle）控制缩放，通过平移构造棱锥的复制体（cube'），同时包含文本对象（texte1、texte2）用于显示公式或提示。④ 适合高中或大学低年级学生，用于立体几何教学、体积公式的探究性学习，也可作为微积分中定积分应用（如旋转体体积）的直观引入。

GeoGebra 命令及参数

  1. Sequence(Mat * Element[cube, k], k, 1, 8) → cube'
  2. Element(Element[AE, 3], 1) → droite
  3. Sequence((Element[cube', k, 1, 1], Element[cube', k, 2, 1]), k, 1, 8) → listepoints
  4. Point(listepoints) → A
  5. Point(listepoints) → E
  6. Point(listepoints) → C_1
  7. Point(listepoints) → D
  8. Polygon(A, E, C_1, D) → poly1, a_1, b, c_1, d
  9. Point(listepoints) → B
  10. Point(listepoints) → F
  11. Point(listepoints) → G
  12. Point(listepoints) → H
  13. Polygon(B, F, G, H) → poly2, e, f, g, h
  14. Point(listepoints) → C
  15. Polygon(F, C, A, B) → poly3, f_1, i, a_2, b_1
  16. Element(AB, 3, 1) → avant
  17. Polygon(C_1, G, H, E) → poly4, c_2, g_1, h_1, e_1
  18. Element(AC, 3, 1) → dessus
  19. Polygon(F, C, C_1, G) → poly5, f_2, c_3, c_4, g_2
  20. Polygon(A, B, H, E) → poly6, a_3, b_2, h_2, e_2
  21. Segment(E, A) → c
  22. Segment(A, C) → j
  23. Segment(A, B) → k
  24. Segment(C, C_1) → l
  25. Segment(C_1, E) → m
  26. Segment(H, E) → n
  27. Segment(A, B) → p
  28. Segment(B, H) → q
  29. Segment(H, E) → r
  30. Segment(H, G) → s
  31. Segment(B, F) → t
  32. Segment(G, C_1) → d_1
  33. Segment(F, G) → i_1
  34. Segment(F, C) → j_1
  35. PointIn(poly5) → J
  36. Translate(J, Vector[val1 * Vector[A, B]]) → pyr1_1
  37. Translate(A, Vector[val1 * Vector[A, B]]) → pyr1_2
  38. Translate(B, Vector[val1 * Vector[A, B]]) → pyr1_3
  39. Translate(H, Vector[val1 * Vector[A, B]]) → pyr1_4
  40. Translate(E, Vector[val1 * Vector[A, B]]) → pyr1_5
  41. Polygon(pyr1_1, pyr1_2, pyr1_3) → poly11, pyr1_6, pyr1_7, pyr1_8
  42. Polygon(pyr1_1, pyr1_3, pyr1_4) → poly12, pyr1_9, pyr1_{10}, pyr1_{11}
  43. Polygon(pyr1_5, pyr1_1, pyr1_2) → poly13, pyr1_{12}, pyr1_{13}, pyr1_{14}
  44. Polygon(pyr1_5, pyr1_4, pyr1_3, pyr1_2) → poly14, pyr1_{15}, pyr1_{16}, pyr1_{17}, pyr1_{18}
  45. Translate(J, Vector[val2 * Vector[A, E]]) → pyr2_1
  46. Translate(E, Vector[val2 * Vector[A, E]]) → pyr2_2
  47. Translate(H, Vector[val2 * Vector[A, E]]) → pyr2_3
  48. Translate(C_1, Vector[val2 * Vector[A, E]]) → pyr2_4
  49. Translate(G, Vector[val2 * Vector[A, E]]) → pyr2_5
  50. Polygon(pyr2_1, pyr2_2, pyr2_3) → poly15, pyr2_6, pyr2_7, pyr2_8
  51. Polygon(pyr2_1, pyr2_4, pyr2_5) → poly16, pyr2_9, pyr2_{10}, pyr2_{11}
  52. Polygon(pyr2_1, pyr2_3, pyr2_5) → poly17, pyr2_{12}, pyr2_{13}, pyr2_{14}
  53. Polygon(pyr2_4, pyr2_2, pyr2_3, pyr2_5) → poly18, pyr2_{15}, pyr2_{16}, pyr2_{17}, pyr2_{18}
  54. Translate(J, Vector[val3 * Vector[D, A]]) → pyr3_1
  55. Translate(F, Vector[val3 * Vector[D, A]]) → pyr3_2
  56. Translate(B, Vector[val3 * Vector[D, A]]) → pyr3_3
  57. Translate(G, Vector[val3 * Vector[D, A]]) → pyr3_4
  58. Translate(H, Vector[val3 * Vector[D, A]]) → pyr3_5
  59. Polygon(pyr3_1, pyr3_3, pyr3_2) → poly19, pyr3_6, pyr3_7, pyr3_8
  60. Polygon(pyr3_2, pyr3_4, pyr3_5, pyr3_3) → poly20, pyr3_9, pyr3_{10}, pyr3_{11}, pyr3_{12}
  61. Polygon(pyr3_1, pyr3_2, pyr3_4) → poly21, pyr3_{13}, pyr3_{14}, pyr3_{15}
  62. Polygon(pyr3_1, pyr3_5, pyr3_4) → poly22, pyr3_{16}, pyr3_{17}, pyr3_{18}