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Cavalieri's Principle by Irina Boyadzhiev

Cavalieri's Principle by Irina Boyadzhiev
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2016-04-14 10:32
Solid GeometryCalculusGeometric Construction
In collections Area, Surface Area, Volume, Cross Section, 3D
This example illustrates Cavalieri's Principle, which states that if two solids have equal heights and the same cross-sectional area at every level, then the two solids have the same volume. Key concepts include solid geometry and calculus. Using GeoGebra commands such as Polygon, Prism, Surface, etc., the construction process of the solid figures is dynamically demonstrated. Suitable for high school and above students learning solid geometry and 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'
  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