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Proof of Sphere Volume

Proof of Sphere Volume
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2019-02-18 10:43
Solid GeometryGeometric Construction
This example demonstrates the proof of the volume of a sphere. Key mathematical concepts include sphere, volume, and special points. Using GeoGebra commands such as Intersect, Circle, Sphere, etc., the calculation method of the sphere's volume is clearly shown. Suitable for high school mathematics education and learning.

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