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Determining Volume by Decomposition

Determining Volume by Decomposition
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2024-09-05 16:07
Solid GeometryTransformationGeometric Construction
In collections Rauminhalt zusammengesetzter Körper - Quader und Würfel
This example demonstrates how to calculate the volume of a three-dimensional shape by decomposing it. Key mathematical concepts include volume computation, geometric transformations, and spatial geometry. Construction involves GeoGebra commands such as Translate, If, Element, and others. Suitable for middle school geometry studies or mathematical modeling exercises.

GeoGebra Commands & Parameters

  1. Translate({Prism[Polygon[(-0.5, 1.5, -1), (-0.5, 1.5, -1) + (3, 0), (-0.5, 1.5, -1) + (3, -3), (-0.5, 1.5, -1) + (0, -3)], (-0.5, 1.5, 5)]}, Vector[(dx / 2 + 0.001, 0, 0)]) → object122HintSenkrecht1
  2. Translate({Prism[Polygon[(-0.5, 1.5), (-0.5, 1.5) + (3, 0), (-0.5, 1.5) + (3, 0, 2), (-0.5, 1.5) + (0, 0, 2)], (-0.5, -1.5)]}, Vector[(0, 0, ((-dz)) / 2 - 1)]) → object122HintWaagrecht1
  3. Translate({Prism[Polygon[(-2.5, 1.5, 1), (-2.5, 1.5, 1) + (5, 0), (-2.5, 1.5, 1) + (5, 0, 4), (-2.5, 1.5, 1) + (0, 0, 4)], (-2.5, -1.5, 1)]}, Vector[(dx, 0, dz / 2 + 0.001)]) → object122HintWaagrecht2
  4. Translate({Prism[Polygon[(-0.5, 1.5, 1), (-0.5, 1.5, 1) + (0, -3), (-0.5, 1.5, 1) + (-2, -3), (-0.5, 1.5, 1) + (-2, 0)], (-0.5, 1.5, 5)]}, Vector[(((-dx)) / 2, 0, 0)]) → object122HintSenkrecht2
  5. If(SelectedIndex[l1] ≟ 2, Volume[object122HintWaagrecht1(1)], Volume[object122HintSenkrecht1(1)]) → sol2
  6. If(SelectedIndex[l1] ≟ 2, Volume[object122HintWaagrecht2(1)], Volume[object122HintSenkrecht2(1)]) → sol1
  7. Element(randomQuestions, Mod[questionCount, Length[randomQuestions]] + 1) → randomQuestion
  8. Element(allQuestions, randomQuestion) → actualQuestion
  9. Shuffle(Sequence[Element[actualQuestion, i], i, 2, numberOfAnswers + 1]) → actualAnswersShuffled, {?, ?, ?, ?, ?, ?}
  10. Element(actualAnswersShuffled, 3) → actualAnswer03
  11. Element(actualAnswersShuffled, 2) → actualAnswer02
  12. Element(actualAnswersShuffled, 1) → actualAnswer01
  13. Element(actualAnswersShuffled, 4) → actualAnswer04
  14. Element(actualAnswersShuffled, 5) → actualAnswer05
  15. Element(actualAnswersShuffled, 6) → actualAnswer06
  16. Element(actualAnswer06, 2) → chk06Expected
  17. Element(actualAnswer05, 2) → chk05Expected
  18. Element(actualAnswer04, 2) → chk04Expected
  19. Element(actualAnswer01, 2) → chk01Expected
  20. Element(actualAnswer02, 2) → chk02Expected
  21. Element(actualAnswer03, 2) → chk03Expected
  22. Cone((0, 0, 2), (0, 0, 5), 1) → figur022kegel, figur012kegel_{2}, figur012kegel_{3}
  23. Cone(((-2.7755575615628914E-17), 0, 1), ((-2.7755575615628914E-17), 0, -3), 1.5) → figur031kegel, figur021kegel_{2}, figur021kegel_{3}
  24. Cone((0, 0, 1), (0, 0, 4), 2.5) → figur042kegel, figur032kegel_{2}, figur032kegel_{3}
  25. Cone((0, 0, 1), (0, 0, 5), 1.5) → figur062kegel, figur062kegel_{2}, figur062kegel_{3}
  26. Textfield(input1) → InputBox1
  27. LaTeX(randomQuestion, true, true) → text1
  28. Remove(l1, {SelectedElement[l1]}) → l2
  29. Textfield(input2) → InputBox2
  30. LaTeX(page, true, true) → text5
  31. Sum({attemptCountInputBox1, attemptCountInputBox2}) → attemptCount
  32. Textfield(input3) → InputBox3
  33. If(page ≟ 2, page2CheckboxChanged, page ≟ 7, SelectedIndex[l2] ≠ 1, true, page ≟ 3, dz ≥ 2 ∨ dx ≥ 2, page ≟ 4, correct, page ≟ 5, dz ≟ 0 ∧ dx ≟ 0, page ≟ 6, correctInputBox3) → fixed
  34. CornerThreeD(-1, 11) → viewActual
  35. SelectedIndex(l1Page8) → selectedIndexPage8
  36. SelectedIndex(l1) → a

AI Drawing Prompt Reference

Visualize a 3D shape decomposition volume calculation example, combining multiple prisms and cones through translation transformations to form a large volume, with labeled volume formulas and key steps, demonstrating geometric transformations and spatial geometry relationships, suitable for educational purposes.
The prompt above was compiled from the original drawing commands and is for reference only.