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Cavalieri's Principle - Volume

Cavalieri's Principle - Volume
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2019-01-13 00:43
Coordinate GeometrySolid GeometryGeometric Construction
This example illustrates Cavalieri's Principle, which states that if two solid figures have equal cross-sectional areas at every level, then they have the same volume. Key concepts include solid geometry and equal areas. Using commands like Polygon, Prism, and Height in GeoGebra, this example provides a visual demonstration of the principle. Suitable for middle school and above students studying solid geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a_1, b
  2. Prism(poly1, 3.245704115496874) → d, D, edgeCF, edgeDE, edgeEF, edgeDF, E, F, faceABED, faceBCFE, faceACFD, faceDEF, edgeAD, edgeBE
  3. Height(d) → h
  4. If(t < 0.5, 0, (2 * (t - 0.5))) → t2
  5. If(t < 0.5, (2 * t), 1) → t1
  6. Polygon(C', A', B') → poly2, b', c', a'
  7. CurveCartesian(f(u) + x1, y1, u, u, 0, h) → e
  8. CurveCartesian(f(u) + x2, y2, u, u, 0, h) → g
  9. CurveCartesian(f(u) + x3, y3, u, u, 0, h) → i
  10. Surface((u * (f(v) + x1)) + ((1 - u) * (f(v) + x2)), (u * y1) + ((1 - u) * y2), v, u, 0, 1, v, 0, h) → j
  11. Surface((u * (f(v) + x1)) + ((1 - u) * (f(v) + x3)), (u * y1) + ((1 - u) * y3), v, u, 0, 1, v, 0, h) → a
  12. Surface((u * (f(v) + x2)) + ((1 - u) * (f(v) + x3)), (u * y2) + ((1 - u) * y3), v, u, 0, 1, v, 0, h) → k
  13. Segment(G, G + (0, 0, h)) → l
  14. Point(l) → H
  15. Polygon(I, J, K) → poly3, k_1, i_1, j_1
  16. Polygon(L, M, N) → poly4, n, l_1, m
  17. Polygon(O, P, Q) → poly5, q, o, p
  18. Polygon((-7, -7, h'), (7, -7, h'), (7, 7, h'), (-7, 7, h')) → poly6, r, s, b_1, c_1
  19. Segment(A, B) → d_1
  20. Segment(B, C) → e_1
  21. Segment(C, A) → f_1
  22. Sequence(Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ))], δ], i, 0, w - 1) → list2
  23. Sequence(Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ))], δ], i, 0, w - 1) → list2'
  24. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, w - 1) → list1
  25. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, w - 1) → list1'
  26. Vector(R0, B) → v
  27. Length(list2) → ll
  28. Sequence(Vector[(f((i * δ)) + x1, y1, (i * δ)), A + (0, 0, (i * δ))], i, 0, w - 1) → list3
  29. Translate(Element[list2, floor(u)], Vector[((u - floor(u)) * Element[list3, floor(u)])]) → h_1
  30. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, floor(u) - 2) → list6
  31. Sequence(Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ))], δ], i, floor(u), w - 1) → list5