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Pyramid and Cube Volume Proof

Pyramid and Cube Volume Proof
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2015-12-03 23:00
Solid GeometryTransformationGeometric Construction
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. Key math concepts include volume calculation, geometric transformations, and spatial visualization. GeoGebra commands and objects like Min, Max, Pyramid, Line, Rotate, Segment, Plane are utilized in the construction. Suitable for high school solid geometry teaching and geometric proof comprehension.

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