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Volume Measurement 4

Volume Measurement 4
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2016-05-30 17:45
Solid GeometryCalculusComplex NumberOptimization
This example demonstrates how to measure and visualize the volume of a cube using GeoGebra. Core mathematical concepts include volume calculation, power functions, rhombuses, complex numbers, optimization problems, and integration. Construction highlights from the summary involve commands like Point, Cube, and Translate, along with objects such as points and faces. Suitable for high school and above students to learn and understand applications of solid geometry, calculus, and optimization problems.

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(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 Drawing Prompt Reference

Draw a cube and visualize its volume measurement. Start by plotting a point, then draw a circle based on this point and obtain another point on the circle. Construct a cube using these two points as vertices. To visualize the volume, translate the cube in different directions, adjusting its faces and edges, to form a 3D representation showing the volume measurement of the cube.
The prompt above was compiled from the original drawing commands and is for reference only.