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Sphere Volume Calculation Demo

Sphere Volume Calculation Demo
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2018-10-23 20:54
Solid GeometryGeometric Construction
This example dynamically demonstrates the volume relationships between a sphere, cylinder, and cone, visually showing the derivation of the sphere volume formula. Core math concepts include solid geometry, volume calculation, geometric transformations, and integral concepts. GeoGebra commands like Circle, Sphere, Cylinder, Cone, Rotate, and Intersect are used in construction. Suitable for high school solid geometry teaching and introductory mathematical modeling.

GeoGebra Commands & Parameters

  1. Circle(A, 1) → c
  2. Circle(B, 1) → d
  3. Point(c) → C
  4. Sphere(A, C) → a
  5. Cylinder(B, D, 1) → b, e, f, g
  6. Cone(D, B, 1) → h, k, i
  7. Segment(E, E + (0, 0, 1)) → j
  8. Point(j) → F
  9. OrthogonalPlane(F, j) → l
  10. IntersectPath(l, a) → p
  11. IntersectPath(l, h) → q
  12. IntersectPath(l, b) → r
  13. Point(q) → H
  14. Rotate(H, α, G) → H'
  15. Intersect(Ray[J, J + (1, 0, 0)], p) → K, N
  16. Segment(A, J) → s
  17. Segment(J, K) → t
  18. Segment(K, A) → a_1
  19. Intersect(Ray[G, G + (1, 0, 0)], q) → L, M
  20. Segment(B, G) → b_1
  21. Segment(G, L) → c_1
  22. Circle(s, C) → d_1
  23. Angle(A, J, K) → β
  24. Angle(B, G, L) → γ
  25. Circle(B, 1, Segment[B, B + (0, 0, 1)]) → e_1
  26. Surface((sqrt(1 - h^(2)) * cos(θ)) + x(A), (sqrt(1 - h^(2)) * sin(θ)) + y(A), h, θ, 0, 6.28319, h, 0, 1) → o
  27. Intersect(Ray[G, H], r) → I,
  28. Rotate(I, α, G) → I'
  29. Polygon(H, H', I', I) → poly1, h_1, h', i', i_1
  30. Sequence(Rotate[poly1, (k * α), G], k, 0, n) → donutSurface