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Pyramid and Cube Volume Proof by Matthias Hornof

Pyramid and Cube Volume Proof by Matthias Hornof
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2014-12-27 21:02
Solid GeometryTransformationGeometric Construction
In collections Surface Area and Volume
This example demonstrates the proof that the volume of a pyramid is one-third the volume of a cube with the same base and height using shearing and rotation transformations. Key math concepts include volume calculation, spatial transformations, and vectors. GeoGebra commands like Min, Max, Pyramid, Line, Rotate, Segment, and Plane are used to construct the geometric figures and transformation paths. Suitable for high school solid geometry teaching and introductory mathematical modeling.

GeoGebra Commands & Parameters

  1. Min((Max[t - 4, 0] * 2), 2) → b_3
  2. Min((Max[t - 3, 0] * 2), 2) → b_2
  3. Max(0, Min[t - 1, 1]) → b_1
  4. Pyramid(A_1, A_2, A_3, A_4, A_5) → a, Fläche5, Kante3, Kante5, Kante7, Kante8, Fläche1, Fläche3, Fläche4, Fläche2, Kante1, Kante4, Kante6, Kante2
  5. Pyramid(B_1, B_2, B_3, B_4, B_5) → c_1, Fläche6, Kante13, Kante14, Kante15, Kante16, Fläche7, Fläche8, Fläche9, Fläche10, Kante9, Kante10, Kante11, Kante12
  6. Pyramid(C_1, C_4, C_5, C_6, C_3) → e, Fläche11, Kante21, Kante22, Kante23, Kante24, Fläche12, Fläche13, Fläche14, Fläche15, Kante17, Kante18, Kante19, Kante20
  7. Min((Max[t - 2, 0] * π / 2), 1.5707963267948966) → dw_3
  8. Line(D, D_1) → s_x
  9. Rotate(e, (-dw_3), s_x) → e'
  10. Segment(A_5, Fp_1) → lot_1
  11. Segment(B_5, Fp_2) → lot_2
  12. Segment(Fp_3, Fp_3 + (0, 0, 4)) → lot_3
  13. Line((2, 0, 0), (2, 1, 0)) → s_{xy1}
  14. Line((-2, 0, 0), (-2, 1, 0)) → s_{xy2}
  15. Segment(A, B) → b
  16. Plane((0, 0, 4), (1, 0, 4), (1, 1, 4)) → c

AI Drawing Prompt Reference

Draw a geometric figure consisting of a pyramid and a cube to demonstrate that the pyramid's volume is one-third of the cube's volume. The figure should include a cube and a pyramid with the same height and base. The construction order is to first draw the cube, then the pyramid, and finally apply shearing and rotation transformations to show the volume relationship. Ensure the figure is clear and labeled to help understand the volume ratio.
The prompt above was compiled from the original drawing commands and is for reference only.