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Volume of pyramid is 1/3 volume of cube (Method II)

Volume of pyramid is 1/3 volume of cube (Method II)
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2020-07-21 21:22
VectorSolid GeometryGeometric Construction
This example demonstrates the relationship between the volume of a pyramid and a cube, specifically that the volume of a pyramid is 1/3 that of a cube with the same base and height. Core mathematical concepts include volume formulas in solid geometry, 3D vectors, and geometric transformations. The construction uses GeoGebra commands/objects like Point, Segment, Cube, Pyramid, Rotate. Suitable for high school students learning solid geometry volume calculations and spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Point(Circle[A, Increase]) → B
  2. Segment(A, B) → f
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → b, D, faceCDHG_1, faceEFGH, edgeAD_1, edgeAB_1, edgeBC_1, edgeCD_1, edgeAE_1, edgeDH_1, edgeEH_1, edgeEF_1, E, edgeBF_1, edgeFG_1, edgeCG_1, edgeGH_1, F, G, H, faceABCD_1, faceADHE, faceABFE, faceBCGF_1
  5. Pyramid(A, B, C, D, E) → c, faceABCD, edgeAE, edgeBE_1, edgeCE_1, edgeDE_1, faceABE, faceBCE_1, faceCDE_1, faceADE, edgeAB, edgeBC_2, edgeCD_2, edgeAD
  6. Pyramid(B, C, G, F, E) → d, faceBCGF, edgeBE, edgeCE_2, edgeEG_1, edgeEF, faceBCE, faceCEG_1, faceEFG, faceBEF, edgeBC, edgeCG_2, edgeFG, edgeBF
  7. Segment(B, C) → g
  8. Rotate(d, Fold, g) → d'
  9. Pyramid(C, D, H, G, E) → h, faceCDHG, edgeCE, edgeDE, edgeEH, edgeEG, faceCDE, faceDEH, faceEGH, faceCEG, edgeCD, edgeDH, edgeGH, edgeCG
  10. Segment(C, D) → i
  11. Rotate(h, Fold, i) → h'

AI Drawing Prompt Reference

Draw a cube with one vertex as the origin and three adjacent edges defining the base square. Inside the cube, construct a pyramid using the four base vertices and the top vertex of the cube, so its base coincides with the cube's bottom face and its apex is at the cube's top. Then construct a second pyramid on another side face of the cube, and rotate it about that side's bottom edge to make it symmetric with the first. Finally, construct a third pyramid on the remaining side face, rotating it about its bottom edge as well. The three pyramids together fill the cube, visually demonstrating that each pyramid's volume is one-third of the cube's volume.
The prompt above was compiled from the original drawing commands and is for reference only.