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Cavalieri's Principle - Volume of a Sphere

Cavalieri's Principle - Volume of a Sphere
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2018-07-29 16:20
FunctionTrigonometrySolid GeometryCalculusGeometric Construction
This example demonstrates the calculation of a sphere's volume using Cavalieri's principle by comparing cross-sectional areas of a cylinder, cone, and sphere. Core mathematical concepts include volume calculation, geometric cross-sections, Cavalieri's principle, and analytical geometry. Construction highlights from the summary involve commands like Sphere, Cylinder, Cone, Segment, Point, OrthogonalPlane, IntersectPath, Corner, Intersect, Line, and Circle. Suitable for high school geometry and calculus students to visually understand volume calculation principles.

GeoGebra Commands & Parameters

  1. Sphere(A, B) → a
  2. Cylinder(C, D, 2) → b, c, d, e
  3. Cone(D, E, 2) → f, g, h
  4. Cone(C, E, 2) → i, k, j
  5. Segment(G, F) → l
  6. Point(l) → H
  7. OrthogonalPlane(H, l) → m
  8. IntersectPath(m, a) → p
  9. IntersectPath(m, b) → q
  10. Corner(q) → M, N, O, P
  11. IntersectPath(m, i) → r
  12. IntersectPath(m, f) → s
  13. Corner(s) → I, J, K, L
  14. Segment(C, D) → n
  15. Segment(Q, B) → t
  16. Intersect(t, m) → R
  17. Intersect(n, m) → S
  18. Line(H, S) → a_1
  19. Intersect(a_1, p) → T, U
  20. Segment(R, T) → b_1
  21. Segment(R, A) → c_1
  22. Segment(A, T) → d_1
  23. Intersect(a_1, q) → V, W
  24. Intersect(a_1, r) → Z, A_1
  25. Segment(Z, E) → e_1
  26. Segment(E, S) → f_1
  27. Segment(S, V) → g_1
  28. Circle(c_1, T) → h_1
  29. Circle(f_1, V) → k_1
  30. Intersect(g_1, r) → B_1,
  31. Circle(n, B_1) → p_1