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Proof of the Volume of a Pyramid

Proof of the Volume of a Pyramid
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2017-05-27 18:07
Solid GeometryGeometric Construction
This example demonstrates the proof of the volume formula for a pyramid. Key mathematical concepts include pyramids, volume, and solid geometry. It uses commands like Polygon, Pyramid, Segment, and Angle in GeoGebra for construction. Suitable for high school mathematics education and students learning solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → A
  2. Point(xAxis) → B
  3. Point(yAxis) → D
  4. Polygon(A, B, C, D) → poly1, a, b, c, d
  5. Pyramid(poly1, -2) → e, E, faceABE, faceBCE, faceCDE, faceADE, edgeAE, edgeBE, edgeCE, edgeDE
  6. Point(zAxis) → G
  7. Polygon(A, G, F, B) → poly2, a_1, g, f, b_1
  8. Pyramid(poly2, -2) → h, H, faceAGH, faceFGH, faceBFH, faceABH, edgeAH, edgeGH, edgeFH, edgeBH
  9. Polygon(B, C, I, F) → poly3, b_2, c_1, i_1, f_1
  10. Pyramid(poly3, 2) → j, J, faceBCJ, faceCIJ, faceFIJ, faceBFJ, edgeBJ, edgeCJ, edgeIJ, edgeFJ
  11. Polygon(C, D, K, I) → poly4, c_2, d_1, k, i_2
  12. Pyramid(poly4, 2) → l, L, faceCDL, faceDKL, faceIKL, faceCIL, edgeCL, edgeDL, edgeKL, edgeIL
  13. Polygon(D, A, G, K) → poly5, d_2, a_2, g_1, k_1
  14. Pyramid(poly5, 2) → m, M, faceADM, faceAGM, faceGKM, faceDKM, edgeDM, edgeAM, edgeGM, edgeKM
  15. Intersect(zAxis, g_1) → N
  16. Polygon(G, F, I, K) → poly6, g_2, f_2, i_3, k_2
  17. Pyramid(poly6, (-(-2))) → n, O, faceFGO, faceFIO, faceIKO, faceGKO, edgeGO, edgeFO, edgeIO, edgeKO
  18. Segment(D, C) → p
  19. Segment(B, F) → q
  20. Segment(I, C) → r
  21. Segment(I, K) → s
  22. Segment(K, D) → t
  23. Segment(K, G) → h_1
  24. Segment(G, F) → j_1
  25. Segment(D, A) → l_1
  26. Intersect(yAxis, a_2) → P
  27. Segment(B, P) → m_1
  28. Segment(I, F) → p_1
  29. Segment(G, K) → q_1
  30. Segment(P, G) → n_1
  31. Segment(Q, E) → r_1
  32. Segment(Q, R) → s_1
  33. Angle(r_1, s_1) → α