MathPlotAI Logo MathPlotAI

Determine Volume by Decomposition Using GeoGebra

Determine Volume by Decomposition Using GeoGebra
Loading the math board and drawing, please wait…
2023-03-16 17:06
Solid GeometryGeometric Construction
In collections Ⓔ Volumen zusammengesetzter Körper aus Quader und Würfel
This example demonstrates how to calculate the volume of a composite solid by decomposing it using GeoGebra tools. Key mathematical concepts include solid geometry volume calculation, decomposition principles, and application of unit cubes. Construction involves using commands like Translate, If, Element, Shuffle, Sum, and CornerThreeD to build 3D models, with objects such as dz, dx, and object122 for spatial transformations. Suitable for elementary and middle school students to understand volume computation methods, applicable in classroom teaching or self-study.

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

Draw a 3D solid decomposition volume illustration: start with a prism-like main body, then create several pieces by translating/copying it so the blocks represent a partition of the solid. Arrange one set of blocks with horizontal offsets to form the composite structure, and arrange another set with vertical offsets to reflect different heights. Set the camera so the full shape and the spatial relationships between the pieces are clearly visible. In the same scene, add several cone solids as comparison objects: their apexes lie on the same vertical axis, with different heights and radii, supporting an intuitive understanding of volume calculation.
The prompt above was compiled from the original drawing commands and is for reference only.