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Volume of a Pyramid - Cavalieri's Principle

Volume of a Pyramid - Cavalieri's Principle
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2015-05-26 19:23
Solid GeometryGeometric Construction
This example demonstrates Cavalieri's Principle applied to the volume of two pyramids with congruent bases and equal heights. The key mathematical concepts involve volume, similarity ratios, and geometric construction. The construction uses GeoGebra commands such as Pyramid, Plane, Segment, Angle, Intersect, and Polygon. It is suitable for high school students or college undergraduates to understand solid geometry and volume calculations.

GeoGebra Commands & Parameters

  1. Pyramid((-10, 0, 0), (-2, 0, 0), (-2, 8, 0), (-10, 8, 0), (-10, 0, 8)) → e, face1, edge5, edge6, edge7, edge8, face2, face3, face4, face5, edge1, edge2, edge3, edge4
  2. Pyramid((0, 0, 0), (8, 0, 0), (8, 8, 0), (0, 8, 0), (4, 4, 8)) → a, face6, edge13, edge14, edge15, edge16, face7, face8, face9, face10, edge9, edge10, edge11, edge12
  3. Plane((0, 0, h), (5, 5, h), (5, (-5), h)) → d
  4. Segment((4, 4, 0), (4, 4, 8)) → b
  5. Segment((4, 4, 0), (8, 8, 0)) → c
  6. Angle((-2, 0, 0), (-10, 0, 0), (-10, 0, 8)) → α
  7. Angle((-10, 8, 0), (-10, 0, 0), (-10, 0, 8)) → β
  8. Angle((8, 8, 0), (4, 4, 0), (4, 4, 8)) → γ
  9. Intersect(edge14, d) → A
  10. Intersect(edge15, d) → B
  11. Intersect(edge16, d) → C
  12. Intersect(edge13, d) → D
  13. Intersect(edge6, d) → E
  14. Intersect(edge7, d) → F
  15. Intersect(edge8, d) → G
  16. Intersect(edge5, d) → H
  17. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  18. Polygon(E, F, G, H) → poly2, e_1, f, g, h_1

AI Drawing Prompt Reference

Draw two pyramids with congruent bases and equal heights. The first pyramid has its base on the x-axis from -10 to -2 and on the y-axis from 0 to 8, with a height of 8. The second pyramid has its base on the x-axis from 0 to 8 and on the y-axis from 0 to 8, also with a height of 8. Draw a plane through points (0,0,h), (5,5,h), and (5,-5,h). Then, draw vertical segments from the vertices of the pyramids to intersect the plane, obtaining four points. Connect these points to form a quadrilateral. Finally, draw two quadrilaterals representing the intersections of the pyramid sides with the plane.
The prompt above was compiled from the original drawing commands and is for reference only.