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Sphere Volume Proof

Sphere Volume Proof
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2015-05-29 08:59
TrigonometrySolid GeometryGeometric Construction
This example dynamically demonstrates and geometrically constructs the derivation of the sphere volume formula. It involves core mathematical concepts such as sphere, volume, similarity transformation, geometric construction, and orthogonal lines. GeoGebra commands and objects used include Intersect, Point, Circle, Distance, Sphere, Cylinder, OrthogonalLine, Segment, IntersectPath, Surface, Polygon, and If. Suitable for high school solid geometry teaching, helping students understand the geometric meaning and derivation logic of the volume formula.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Distance(A, B) → r
  5. Point(xAxis) → D
  6. Circle(D, r) → h
  7. Sphere(A, r) → a
  8. Cylinder(h, r) → b, d, e
  9. Cylinder(h, (-r)) → f, g, i
  10. Point(xAxis) → C
  11. OrthogonalLine(C, xOyPlane) → j
  12. Segment(E, C) → k
  13. Point(k) → G
  14. IntersectPath(l, a) → p
  15. IntersectPath(l, b) → q
  16. Surface((u * cos(θ)) + Distance[A, D], (u * sin(θ)), u, θ, 0, 6.283185307179586, u, (-r), r) → m
  17. IntersectPath(l, f) → s
  18. OrthogonalLine(D, l) → n
  19. Surface(x(H) + (u * cos(θ)), (u * sin(θ)), z(G), u, z(G), r, θ, 0, 6.283185307179586) → o
  20. Segment(A, B) → t
  21. Intersect(h, xAxis, 2) → J
  22. Segment(D, J) → a_1
  23. Segment(A, I) → b_1
  24. Segment(D, H) → c_1
  25. Intersect(u, p) → K, L
  26. Segment(I, L) → d_1
  27. Segment(A, L) → e_1
  28. Segment(H, N) → f_1
  29. Segment(C, G) → g_1
  30. Segment(A, L) → h_1
  31. Polygon(A, I, L) → poly1, l_1, a_2, i_1
  32. Segment(S, R) → j_1
  33. Polygon(S, R, D) → poly2, d_2, s_1, r_1
  34. Segment(D, S) → k_1
  35. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → v
  36. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → w
  37. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → m_1
  38. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → n_1
  39. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → o_1
  40. If(Step ≠ 10, Step ≟ Step + 1, Step ≟ 0) → p_1