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Demonstrating Sphere Volume Formula by Anthony OR 柯志明

Demonstrating Sphere Volume Formula by Anthony OR 柯志明
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2014-10-04 16:41
Solid GeometryCalculusGeometric ConstructionGraphing
In collections Area, Surface Area, Volume, Cross Section, 3D 3D Geometry & Cross Sections Surface Area and Volume Surface Area and Volume Surface Area and Volume Surface Area & Volume Surface Area & Volume Surface Area and Volume Volumes and Surface Areas of Pyramids, Cones and Spheres GA Chapter 11: Surface Area and Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume GCSE Higher Ch7 Area and Volume More about Volumes and Surface Areas Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres
This example visually demonstrates the derivation of the sphere volume formula 4/3 πr³ through dynamic constructions. Key math concepts include sphere geometry, volume calculation, and integral concepts (via cylinder-cone volume relationships). GeoGebra commands like Circle, Sphere, Cylinder, Cone, and Intersect are used to build the geometric model. Suitable for high school solid geometry and introductory calculus students to understand the geometric meaning of the formula.

GeoGebra Commands & Parameters

  1. Circle(A, 1) → c
  2. Circle(B, 1, xOyPlane) → 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. Intersect(Ray[J, J + (1, 0, 0)], p) → K, N
  18. Segment(A, J) → s
  19. Segment(J, K) → t
  20. Segment(K, A) → a_1
  21. Intersect(Ray[G, G + (1, 0, 0)], q) → L, M
  22. Segment(B, G) → b_1
  23. Segment(G, L) → c_1
  24. Circle(s, C) → d_1
  25. Angle(A, J, K) → β
  26. Angle(B, G, L) → γ
  27. 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
  28. Surface(x(B) + (k * cos(θ)), y(B) + (k * sin(θ)), s, θ, 0, (2 * pi), k, s, 1) → m

AI Drawing Prompt Reference

Draw a sphere with center A and radius 1. Then draw a cylinder and a cone, both with base radius 1, such that their bases coincide with the sphere's base. Use rotation and translation to demonstrate the derivation of the sphere volume formula 4/3πr³, paying special attention to how the intersection of the cylinder and cone with the sphere forms a circle.
The prompt above was compiled from the original drawing commands and is for reference only.