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Shearing and Rotating Proof: Pyramid Volume Equals One-Third of a Cube

Shearing and Rotating Proof: Pyramid Volume Equals One-Third of a Cube
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2021-11-05 23:05
PolygonSolid GeometryTransformationGeometric Construction
In collections Visual Geometry for Grade 9
This example demonstrates a visual proof of the volume relation for a pyramid: V_pyramid = (1/3)·V_cube, using the idea of shearing and rotating to re-partition and rearrange the solids into corresponding equal-volume pieces. Key math concepts include solid geometry volume, volume conservation under geometric transformations, and proportional reasoning that explains the specific factor 1/3. Construction highlights: Min/Max are used to control parameter ranges; Pyramid creates the pyramid solid; Plane defines slicing/interaction surfaces; Line and Segment establish the controlling edges and connections; Rotate performs the reorientation that matches regions between pyramid and cube. The model relies on multiple Fläche objects (area/measure displays tied to the sliced parts) and Kante objects (edges) to organize the decomposition and highlight invariance during transformation. Suitable for upper-secondary to early university geometry instruction, for teaching solid-geometry volume theorems and supporting exploration-based proofs.

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

Create a 3D geometric illustration showing a pyramid with volume equal to one-third of a cube. The cube is divided into three pyramids, one of which is rotated and sheared to rearrange and match the other parts. Use parameters to control the rotation angle and shearing amount, allowing the pyramid to gradually deform and recombine. Draw line segments for perpendicular distances and key edges, and a plane to show the cutting surface. The scene emphasizes volume conservation during geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.