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Demonstrating Sphere Volume Formula by Dan Dale

Demonstrating Sphere Volume Formula by Dan Dale
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2019-07-01 21:05
Solid GeometryCalculusGeometric Construction
This example dynamically demonstrates how to derive the volume formula of a sphere by rotating a cylinder and a cone. Key math concepts include volume calculation, solids of revolution, integral concepts, and geometric transformations. GeoGebra commands and objects used in construction are Circle, Cylinder, Cone, Rotate, and Surface. Suitable for high school solid geometry teaching to help students visually understand the origin of the sphere's volume formula.

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. Intersect(Ray[G, H], r) → I,
  15. Rotate(H, α, G) → H'
  16. Rotate(I, α, G) → I'
  17. Polygon(H, H', I', I) → poly1, h_1, h', i', i_1
  18. Sequence(Rotate[poly1, (k * α), G], k, 0, n) → list1
  19. Intersect(Ray[J, J + (1, 0, 0)], p) → K,
  20. Segment(A, J) → s
  21. Segment(J, K) → t
  22. Segment(K, A) → a_1
  23. Intersect(Ray[G, G + (1, 0, 0)], q) → L, M
  24. Segment(B, G) → b_1
  25. Segment(G, L) → c_1
  26. Circle(s, C) → d_1
  27. Angle(A, J, K) → β
  28. Angle(B, G, L) → γ
  29. Circle(B, 1, Segment[B, B + (0, 0, 1)]) → e_1
  30. 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