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Dividing a Cuboid into Three Equal Volume Triangular Pyramids

Dividing a Cuboid into Three Equal Volume Triangular Pyramids
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2015-01-22 17:50
Solid GeometryGeometric Construction
This example demonstrates that any rectangular cuboid can be divided into three triangular-based pyramids of equal volume. It involves core mathematical concepts such as volume calculation, geometric division, and solid geometry. The construction uses commands like Line, Point, Intersect, Polygon, OrthogonalLine, Prism, Pyramid, Sphere, PointIn, Vector, and Translate to create geometric shapes and calculate volumes. It is suitable for high school geometry teaching and geometric construction learning.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → a
  2. Line(A, yAxis) → b
  3. Point(a) → B
  4. Point(b) → D
  5. Line(B, b) → h
  6. Line(D, a) → i
  7. Intersect(h, i) → C
  8. Polygon(A, B, C, D) → base, a_1, b_1, c, d
  9. OrthogonalLine(A, base) → e
  10. Point(e) → E
  11. Prism(base, E) → f, F, edgeBF_1, edgeCG, edgeDH, edgeEF_1, edgeFG_1, edgeGH, edgeEH, G, H, faceABFE, faceBCGF, faceCDHG, faceADHE, faceEFGH, edgeAE
  12. Pyramid(Polygon[A, B, C, D], F) → uno, faceABF, faceBCF, faceCDF_1, faceADF_1, edgeAF_1, edgeBF, edgeCF_1, edgeDF_1
  13. Pyramid(Polygon[D, C, G, H], F) → due, faceCDF, faceCFG, faceFGH, faceDFH_1, edgeDF_2, edgeCF, edgeFG, edgeFH_1
  14. Pyramid(Polygon[A, D, H, E], F) → tre, faceADF, faceDFH, faceEFH, faceAEF, edgeAF, edgeDF, edgeFH, edgeEF
  15. Sphere(I, J) → g
  16. PointIn(g) → K
  17. PointIn(g) → L
  18. PointIn(g) → M
  19. Vector(I, K) → u
  20. Vector(I, L) → v
  21. Vector(I, M) → w
  22. Translate(uno, u) → uno'
  23. Translate(due, v) → due'
  24. Translate(tre, w) → tre'
  25. Volume(uno) → v1
  26. Volume(due) → v2
  27. Volume(tre) → v3
  28. Volume(f) → vp

AI Drawing Prompt Reference

Draw a rectangular cuboid and divide it into three pyramids of equal volume. Start by drawing the cuboid's base, then construct three pyramids using triangular sections of the base as their bases, with their apexes above the center of the cuboid.
The prompt above was compiled from the original drawing commands and is for reference only.