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Cavalieri's Principle Demo

Cavalieri's Principle Demo
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2016-04-11 15:19
Solid GeometryCalculusIntegralGeometric Construction
In collections Area, Surface Area, Volume, Cross Section, 3D Primary School Mathematics Collection Volumes and Surface Areas of Pyramids, Cones and Spheres Mensuration Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes in Ancient Chinese Mathematics 中國古算中的體積 More about Volumes and Surface Areas Volumes of Pyramid and Sphere in Ancient Chinese Mathematics Connolly Geometry Volume
This example dynamically demonstrates Cavalieri's Principle by showing two solids with equal cross-sectional areas. It involves core concepts like volume, cross-sectional area, integral calculus, and solid geometry. GeoGebra commands and objects used include Polygon, Prism, Height, CurveCartesian, Surface, Segment, Point, and Sequence. Suitable for high school students and math enthusiasts learning about solid geometry and integral calculus.

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'