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Volume Measurement 5

Volume Measurement 5
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2016-05-30 17:55
Solid GeometryCalculusOptimization
In collections Geo&Alg:Area Volume Pythagorean Th.
This example demonstrates how to understand volume concepts through constructing and measuring a cube. It involves core math concepts like volume calculation, integration, line segments, linear programming or optimization, statistical features, and piecewise functions. GeoGebra commands and objects like Point, Cube, Translate are used in the construction. Suitable for high school students learning solid geometry and calculus.

GeoGebra Commands & Parameters

  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(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'
  60. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0)]) → a'
  61. 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'
  62. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1)]) → c'
  63. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -2)]) → i'
  64. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -1)]) → d'
  65. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, -2)]) → j'
  66. 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'
  67. 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'
  68. 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'
  69. 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'
  70. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[D]) → b_1
  71. Translate(facez, Vector[(2, 0, 0)]) → facez200
  72. Translate(facex, Vector[(1, 0, 1)]) → facex101
  73. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1, 1)]) → u
  74. Translate(facez, Vector[(2, -1, 0)]) → facez210
  75. Translate(facex, Vector[(1, -1, 1)]) → facex111
  76. Translate(facey, Vector[(2, 0, 1)]) → facey201
  77. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 0, 1)]) → v
  78. Translate(facez, Vector[(2, -2, 0)]) → facez220
  79. Translate(facex, Vector[(1, -2, 1)]) → facex121
  80. Translate(facey, Vector[(2, -1, 1)]) → facey211
  81. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -1, 1)]) → w
  82. Translate(facez, Vector[(2, -3, 0)]) → facez230
  83. Translate(facex, Vector[(1, -3, 1)]) → facex131
  84. Translate(facey, Vector[(2, -2, 1)]) → facey221
  85. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -2, 1)]) → a_1
  86. 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'
  87. 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'
  88. 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'
  89. 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'
  90. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1, 1)]) → u'
  91. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 0, 1)]) → v'
  92. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -1, 1)]) → w'
  93. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -2, 1)]) → a_1'
  94. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1)]) → k'
  95. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 0)]) → l'
  96. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -1)]) → m'
  97. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, -2)]) → n'
  98. Translate(Prism[Polygon[(-5, 2, 0), (-5, -2, 0), (-2, -2, 0), (-2, 2, 0)], (-5, 2, 2)], Vector[(0, 0)]) → c_1
  99. Translate(Prism[Polygon[(-5, 2, 0), (-5, -2, 0), (-2, -2, 0), (-2, 2, 0)], (-5, 2, 2)], Vector[(0, 0)]) → d_1
  100. Translate(Polygon[(-5, 1, 0), (-2, 1, 0), (-2, 1, 2), (-5, 1, 2)], Vector[(0, 0)]) → poly1
  101. Translate(Polygon[(-5, 0, 0), (-2, 0, 0), (-2, 0, 2), (-5, 0, 2)], Vector[(0, 0)]) → poly2
  102. Translate(Polygon[(-5, -1, 0), (-2, -1, 0), (-2, -1, 2), (-5, -1, 2)], Vector[(0, 0)]) → poly3
  103. Translate(Polygon[(-4, 2, 0), (-4, -2, 0), (-4, -2, 2), (-4, 2, 2)], Vector[(0, 0)]) → poly4
  104. Translate(Polygon[(-3, 2, 0), (-3, -2, 0), (-3, -2, 2), (-3, 2, 2)], Vector[(0, 0)]) → poly5
  105. Translate(Polygon[(-5, -2, 1), (-2, -2, 1), (-2, 2, 1), (-5, 2, 1)], Vector[(0, 0)]) → poly6