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Calculating Volume of Composite Solids

Calculating Volume of Composite Solids
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2022-12-02 21:07
Solid GeometryGeometric ConstructionSlider Animation
This example demonstrates how to calculate the volume of a composite solid formed by two rectangular prisms. Key mathematical concepts include volume computation, geometric transformations, and spatial relationships. Construction involves using GeoGebra commands like Point, Sequence, Prism, Translate to build prism models and observe volume changes by adjusting dimensions and orientations. Suitable for elementary school students (grades 4-5) to develop spatial reasoning and volume calculation skills.

GeoGebra Commands & Parameters

  1. Point(Segment[A_{1} + (2, 0, 0), A_{1} + (8, 0, 0)]) → ED_{1}
  2. Sequence(Translate[A_{1}, Vector[(ii, 0, 0)]], ii, 1, x(ED_{1}) + 4 - 1) → l5
  3. Point(Segment[ED_{1} + (0, 1, 0), ED_{1} + (0, 8, 0)]) → FD_{1}
  4. Point(l5) → CD_{1}
  5. Point(Segment[A_{1} + (0, 0, 1), A_{1} + (0, 0, 8)]) → GD_{1}
  6. Zip((x(CD_{1}), y(CD_{1}), kk), kk, Sequence[1, 8]) → l4
  7. Point(l4) → ID_{1}
  8. Polygon({A_{1}, B_{1}, DD_{1}, CD_{1}}) → base1_{1}
  9. Polygon({CD_{1}, ED_{1}, FD_{1}, DD_{1}}) → base2_{1}
  10. Prism(base1_{1}, GD_{1}) → prism1, K_{1}, edge2_{1}, edge3_{1}, edge4_{1}, edgeGDK_{1}, edgeKL_{1}, edgeLM_{1}, edgeGDM_{1}, L_{1}, M_{1}, face1_{1}, face3_{1}, face4_{1}, face2_{1}, faceGDKLM_{1}, edge1_{1}
  11. Prism(base2_{1}, ID_{1}) → prism2, N_{1}, edge6_{1}, edge7_{1}, edge8_{1}, edgeIDN_{1}, edgeNO_{1}, edgeOP_{1}, edgeIDP_{1}, O_{1}, P_{1}, face5_{1}, face7_{1}, face8_{1}, face6_{1}, faceIDNOP_{1}, edge5_{1}
  12. Polygon({A_{1}, ED_{1}, FD_{1}, B_{1}}) → base3_{1}
  13. Prism(base3_{1}, If[z(GD_{1}) ≤ z(ID_{1}), z(GD_{1}), z(ID_{1})]) → prism3, Q_{1}, edge9_{1}, edge10_{1}, edge11_{1}, edge12_{1}, edgeQR_{1}, edgeRS_{1}, edgeST_{1}, edgeQT_{1}, R_{1}, S_{1}, T_{1}, face9_{1}, face11_{1}, face12_{1}, face10_{1}, faceQRST_{1}
  14. If(z(ID_{1}) ≥ z(GD_{1}), Prism[Translate[Polygon[{CD_{1}, ED_{1}, FD_{1}, DD_{1}}], Vector[A_{1}, GD_{1}]], ID_{1}], Prism[Translate[Polygon[{A_{1}, CD_{1}, DD_{1}, B_{1}}], Vector[CD_{1}, ID_{1}]], GD_{1}]) → prism4
  15. Sequence(Translate[A_{1}, Vector[(jj, 0, 0)]], jj, 0, x(CD_{1}) - x(A_{1})) → l1
  16. Translate(l1, Vector[A_{1}, B_{1}]) → l1'
  17. Translate(l1, Vector[A_{1}, GD_{1}]) → l1'_{1}
  18. Zip(Segment[AA, BB], AA, l1, BB, l1') → squaresBase1
  19. Translate(squaresBase1, Vector[A_{1}, GD_{1}]) → squaresBase1'
  20. Zip(Segment[AA, BB], AA, l1, BB, l1'_{1}) → squaresFront1
  21. Translate(squaresFront1, Vector[A_{1}, B_{1}]) → squaresFront1'
  22. Sequence(Translate[A_{1}, Vector[(0, jj, 0)]], jj, 0, y(B_{1}) - y(A_{1})) → l2
  23. Translate(l2, Vector[A_{1}, GD_{1}]) → l2'
  24. Sequence(Translate[A_{1}, Vector[(0, 0, jj)]], jj, 0, z(GD_{1}) - z(A_{1})) → l3
  25. Translate(l3, Vector[A_{1}, B_{1}]) → l3'
  26. Zip(Segment[AA, BB], AA, l3, BB, l3') → squaresLeft1
  27. Zip(Segment[AA, BB], AA, l2, BB, l2') → squaresLeft2
  28. Translate(squaresLeft1, Vector[A_{1}, CD_{1}]) → squaresLeft1'
  29. Translate(squaresLeft2, Vector[A_{1}, CD_{1}]) → squaresLeft2'
  30. Translate(l3, Vector[A_{1}, CD_{1}]) → l3'_{1}
  31. Zip(Segment[AA, BB], AA, l3, BB, l3'_{1}) → squaresBack
  32. Translate(squaresBack, Vector[A_{1}, B_{1}]) → squaresBack'
  33. Translate(l2', Vector[A_{1}, CD_{1}]) → l2''
  34. Zip(Segment[AA, BB], AA, l2', BB, l2'') → squaresBottom
  35. Translate(squaresBottom, Vector[GD_{1}, A_{1}]) → squaresBottom'
  36. Sequence(Sequence[Polygon[CD_{1} + (ii, 0, jj), CD_{1} + (ii + 1, 0, jj), CD_{1} + (ii + 1, 0, jj + 1), CD_{1} + (ii, 0, jj + 1)], ii, 0, x(ED_{1}) - x(CD_{1}) - 1], jj, 0, z(N_{1}) - 1) → squaresFront2, {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}}
  37. Sequence(Sequence[Polygon[ED_{1} + (0, ii, jj), ED_{1} + (0, ii + 1, jj), ED_{1} + (0, ii + 1, jj + 1), ED_{1} + (0, ii, jj + 1)], ii, 0, y(FD_{1}) - y(ED_{1}) - 1], jj, 0, z(N_{1}) - 1) → squaresRight, {{1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}}
  38. Translate(squaresFront2, Vector[ED_{1}, FD_{1}]) → squaresFront2'
  39. Translate(squaresRight, Vector[ED_{1}, CD_{1}]) → squaresRight'
  40. Sequence(Sequence[Polygon[CD_{1} + (ii, jj, 0), CD_{1} + (ii + 1, jj, 0), CD_{1} + (ii + 1, jj + 1, 0), CD_{1} + (ii, jj + 1, 0)], ii, 0, x(ED_{1}) - x(CD_{1}) - 1], jj, 0, y(FD_{1}) - y(ED_{1}) - 1) → squaresBottom2, {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}}
  41. Translate(squaresBottom2, Vector[ED_{1}, N_{1}]) → squaresBottom2'
  42. If(z(GD_{1}) ≥ z(ID_{1}), abs(x(GD_{1}) - x(ID_{1})), abs(x(ID_{1}) - x(ED_{1}))) → base4
  43. Corner(5) → A
  44. Min(z(GD_{1}), z(ID_{1})) → hei3

AI Drawing Prompt Reference

Create a composite solid made of two rectangular prisms connected together in 3D so the overall volume structure and changes in height can be visually compared. First form a base polygon for the first prism on a rectangular footprint, then extend it upward along the vertical reference direction to build the first prism. Construct a second base polygon with the same spatial setup and generate the second prism, aligning it to the first so the two prisms form one combined solid. Make the height follow a rule based on the two end reference points, producing a step-like top surface where one side uses the smaller (or larger) height. Finally, cover key visible faces with small square tiles arranged in directions parallel to the coordinate axes to create a clear grid-like visual measure, keeping the connection edges and height references visible.
The prompt above was compiled from the original drawing commands and is for reference only.