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测量体积4

测量体积4
几何画板及绘图内容加载中，请稍等……
2016-05-30 17:45
立体几何微积分复数最值优化
所属专题 Geo&Alg:Area Volume Pythagorean Th.
①本范例演示了如何通过GeoGebra测量并可视化立方体的体积；②涉及的核心数学概念包括体积计算、幂函数、菱形、复数、最优化问题和积分；③作图要点提炼自构造摘要，使用了Point（点）、Cube（立方体）、Translate（平移）等命令和对象；④适合高中及以上学生学习和理解立体几何、微积分及优化问题的实际应用。

GeoGebra 命令及参数

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Point(Circle[J, Distance[I, J], Segment[I, J]]) → K
  3. Point(Circle[R, Distance[Q, R], Segment[Q, R]]) → S
  4. Point(Circle[H_1, Distance[G_1, H_1], Segment[G_1, H_1]]) → I_1
  5. Cube(G_1, H_1, I_1) → e_1, J_1, face11, facez, edge13, edge12, edge15, edge17, edge14, edge19, edge21, edge20, K_1, edge16, edge22, edge18, edge23, L_1, M_1, N_1, face7, face9, facey, facex
  6. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (2, 1)]) → f
  7. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 0)]) → g
  8. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1, 1)]) → h
  9. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1)]) → e
  10. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1)]) → e'
  11. Translate(facex, Vector[(0, 0, 0), (0, -1, 0)]) → facex010
  12. Translate(facey, Vector[(0, 0), (1, 0, 0)]) → facey100
  13. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0)]) → a
  14. Translate(facez, Vector[(0, 0, 0), (0, -1, 0)]) → facez010
  15. Translate(facey, Vector[(0, 0, 1)]) → facey001
  16. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 1)]) → b
  17. Translate(facey, Vector[(0, -1, 0)]) → facey010
  18. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1)]) → c
  19. Translate(facey, Vector[(1, -1, 0)]) → facey110
  20. Translate(facex, Vector[(0, -2, 0)]) → facex020
  21. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -1)]) → d
  22. Translate(facey, Vector[(0, -2, 0)]) → facey020
  23. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2)]) → i
  24. Translate(facey, Vector[(1, -2, 0)]) → facey120
  25. Translate(facex, Vector[(0, -3, 0)]) → facex030
  26. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -2)]) → j
  27. Translate(facex, Vector[(1, 0, 0)]) → facex100
  28. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1)]) → k
  29. Translate(facex, Vector[(1, -1, 0)]) → facex110
  30. Translate(facey, Vector[(2, 0, 0)]) → facey200
  31. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 0)]) → l
  32. Translate(facey, Vector[(2, -1, 0)]) → facey210
  33. Translate(facex, Vector[(1, -2, 0)]) → facex120
  34. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -1)]) → m
  35. Translate(facey, Vector[(2, -2, 0)]) → facey220
  36. Translate(facex, Vector[(1, -3, 0)]) → facex130
  37. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -2)]) → n
  38. Translate(facey, Vector[(0, -1, 1)]) → facey011
  39. Translate(facez, Vector[(0, -2, 0)]) → facez020
  40. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1, 1)]) → o
  41. Translate(facez, Vector[(0, -3, 0)]) → facez030
  42. Translate(facey, Vector[(0, -2, 1)]) → facey021
  43. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2, 1)]) → p
  44. Translate(facex, Vector[(0, 0, 1)]) → facex001
  45. Translate(facez, Vector[(1, 0, 0)]) → facez100
  46. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 1, 1)]) → q
  47. Translate(facex, Vector[(0, -1, 1)]) → facex011
  48. Translate(facey, Vector[(1, 0, 1)]) → facey101
  49. Translate(facez, Vector[(1, -1, 0)]) → facez110
  50. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0, 1)]) → r
  51. Translate(facex, Vector[(0, -2, 1)]) → facex021
  52. Translate(facey, Vector[(1, -1, 1)]) → facey111
  53. Translate(facez, Vector[(1, -2, 0)]) → facez120
  54. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -1, 1)]) → s
  55. Translate(facex, Vector[(0, -3, 1)]) → facex031
  56. Translate(facey, Vector[(1, -2, 1)]) → facey121
  57. Translate(facez, Vector[(1, -3, 0)]) → facez130
  58. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -2, 1)]) → t
  59. Translate(facez, Vector[(0, 0, 1)]) → facez001
  60. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 1, 2)]) → u
  61. Translate(facey, Vector[(0, 0, 2)]) → facey002
  62. Translate(facez, Vector[(0, -1, 1)]) → facez011
  63. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 2)]) → v
  64. Translate(facez, Vector[(0, -2, 1)]) → facez021
  65. Translate(facey, Vector[(0, -1, 2)]) → facey012
  66. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1, 2)]) → w
  67. Translate(facez, Vector[(0, -3, 1)]) → facez031
  68. Translate(facey, Vector[(0, -2, 2)]) → facey022
  69. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2, 2)]) → a_1
  70. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (2, 1)]) → f'
  71. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0)]) → a'
  72. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 0)]) → l'
  73. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 0)]) → g'
  74. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1)]) → k'
  75. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1)]) → c'
  76. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2)]) → i'
  77. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -1)]) → d'
  78. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -2)]) → j'
  79. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -1)]) → m'
  80. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -2)]) → n'
  81. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1, 1)]) → h'
  82. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 1)]) → b'
  83. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1, 1)]) → o'
  84. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2, 1)]) → p'
  85. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 1, 1)]) → q'
  86. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0, 1)]) → r'
  87. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -1, 1)]) → s'
  88. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -2, 1)]) → t'
  89. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 1, 2)]) → u'
  90. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 2)]) → v'
  91. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1, 2)]) → w'
  92. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2, 2)]) → a_1'
  93. Translate(Prism[Polygon[(1, 2, 0), (4, 2, 0), (4, 2, 1), (3, 2, 1), (3, 2, 2), (2, 2, 2), (2, 2, 3), (1, 2, 3)], (1, -2, 0)], Vector[(-6, 0)]) → b_1
  94. Translate(Prism[Polygon[(1, 2, 0), (4, 2, 0), (4, 2, 1), (3, 2, 1), (3, 2, 2), (2, 2, 2), (2, 2, 3), (1, 2, 3)], (1, -2, 0)], Vector[(-6, 0)]) → c_1
  95. Translate(Polygon[(1, 2, 0), (4, 2, 0), (4, 2, 1), (3, 2, 1), (3, 2, 2), (2, 2, 2), (2, 2, 3), (1, 2, 3)], Vector[(-6, -1)]) → poly1
  96. Translate(Polygon[(1, 2, 0), (4, 2, 0), (4, 2, 1), (3, 2, 1), (3, 2, 2), (2, 2, 2), (2, 2, 3), (1, 2, 3)], Vector[(-6, -2)]) → poly2
  97. Translate(Polygon[(-5, 2, 2), (-5, -2, 2), (-4, -2, 2), (-4, 2, 2)], Vector[(0, 0)]) → poly4
  98. Translate(Polygon[(-5, 2, 1), (-5, -2, 1), (-3, -2, 1), (-3, 2, 1)], Vector[(0, 0)]) → poly5
  99. Translate(Polygon[(1, 2, 0), (4, 2, 0), (4, 2, 1), (3, 2, 1), (3, 2, 2), (2, 2, 2), (2, 2, 3), (1, 2, 3)], Vector[(-6, -3)]) → poly3
  100. Translate(Polygon[(-4, 2, 2), (-4, -2, 2), (-4, -2, 0), (-4, 2, 0)], Vector[(0, 0)]) → poly6
  101. Translate(Polygon[(-3, 2, 1), (-3, -2, 1), (-3, -2, 0), (-3, 2, 0)], Vector[(0, 0)]) → poly7

AI 绘图提示语参考

画一个立方体，并展示其体积测量。首先绘制一个点，然后基于这个点绘制一个圆，并在这个圆上得到另一个点。接着，以这两个点为顶点构建一个立方体。为了可视化体积，将立方体沿不同方向平移，并调整其面和边，最终形成一个展示立方体体积测量的立体图形。
以上提示语为根据原始绘图命令整理而来，仅供参考。