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Find the volume of a composite solid by clever completion in GeoGebra 3D

Find the volume of a composite solid by clever completion in GeoGebra 3D
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2023-03-16 17:06
FunctionSolid GeometryGeometric ConstructionSlider Animation
In collections Ⓔ Volumen zusammengesetzter Körper aus Quader und Würfel
This example demonstrates how to determine the volume of a composite solid in GeoGebra 3D using two complementary ideas: decomposing the body into unit cubes and computing the volume by cleverly completing it into an appropriate cuboid (then accounting for the overlap/removed part equivalently). Key mathematics includes the cuboid volume formula V = length × width × height, the volume-as-counting principle via unit cubes, and the idea of volume invariance under equivalent rearrangements/completions. Construction-wise, the model leverages commands such as Translate and Volume to manage 3D placement and compute volumes, Element/Textfield/LaTeX to present expressions, Sum and If to assemble results and validate answers, and Shuffle/Corner to randomize the question/answers. Objects like listSolution, correctInputBox*, and step lists support guided, step-by-step learning with both multiple-choice and fill-in/open responses. Suitable for grades around 5–6 or early middle-school, especially for interactive practice and formative assessment on solid geometry and volume reasoning.

GeoGebra Commands & Parameters

  1. Translate({Prism[Polygon[(-2.5, -1.5, -1), (-2.5, -1.5, -1) + (2, 0), (-2.5, -1.5, -1) + (2, 3), (-2.5, -1.5, -1) + (0, 3)], 2]}, Vector[(-0.001, 0, -0.001 - dy)]) → object02
  2. Volume(object02(1)) → sol2
  3. 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
  4. Element(randomQuestions, Mod[questionCount, Length[randomQuestions]] + 1) → randomQuestion
  5. Element(allQuestions, randomQuestion) → actualQuestion
  6. Shuffle(Sequence[Element[actualQuestion, i], i, 2, numberOfAnswers + 1]) → actualAnswersShuffled, {?, ?, ?, ?, ?, ?}
  7. Element(actualAnswersShuffled, 3) → actualAnswer03
  8. Element(actualAnswersShuffled, 2) → actualAnswer02
  9. Element(actualAnswersShuffled, 1) → actualAnswer01
  10. Element(actualAnswersShuffled, 4) → actualAnswer04
  11. Element(actualAnswersShuffled, 5) → actualAnswer05
  12. Element(actualAnswersShuffled, 6) → actualAnswer06
  13. Element(actualAnswer06, 2) → chk06Expected
  14. Element(actualAnswer05, 2) → chk05Expected
  15. Element(actualAnswer04, 2) → chk04Expected
  16. Element(actualAnswer01, 2) → chk01Expected
  17. Element(actualAnswer02, 2) → chk02Expected
  18. Element(actualAnswer03, 2) → chk03Expected
  19. Cone((0, 0, 2), (0, 0, 5), 1) → figur022kegel, figur012kegel_{2}, figur012kegel_{3}
  20. Cone(((-2.7755575615628914E-17), 0, 1), ((-2.7755575615628914E-17), 0, -3), 1.5) → figur031kegel, figur021kegel_{2}, figur021kegel_{3}
  21. Cone((0, 0, 1), (0, 0, 4), 2.5) → figur042kegel, figur032kegel_{2}, figur032kegel_{3}
  22. Cone((0, 0, 1), (0, 0, 5), 1.5) → figur062kegel, figur062kegel_{2}, figur062kegel_{3}
  23. Textfield(input1) → InputBox1
  24. LaTeX(randomQuestion, true, true) → text1
  25. Remove(l1, {SelectedElement[l1]}) → l2
  26. Textfield(input2) → InputBox2
  27. LaTeX(page, true, true) → text5
  28. Sum({attemptCountInputBox1, attemptCountInputBox2}) → attemptCount
  29. Textfield(input3) → InputBox3
  30. If(page ≟ 2, showQuader2, page ≟ 8, correctInputBox3, true, page ≟ 4, correctInputBox1, page ≟ 5, dy ≥ 1.78, page ≟ 6, correctInputBox2, page ≟ 7, sliderGrey ≥ 0.97) → fixed
  31. CornerThreeD(-1, 11) → viewActual

AI Drawing Prompt Reference

Draw a composite solid made up of several geometric shapes, including a base and a tall cone. The base consists of multiple stacked unit cubes, and the cone sits atop the base, forming an irregular three-dimensional shape. Use transparency to reveal the internal structure, and label key dimensions such as the length, width, and height of the base, as well as the height of the cone, for volume calculation.
The prompt above was compiled from the original drawing commands and is for reference only.