Friendly Sphere

나윤상, May 13, 2017


  • 1. Volume of Spheres with Proof
  • 2. Sphere Slider
  • 3. Are there dilation on a sphere?
  • 4. A spiral on a sphere
  • 5. Surface Area of Spheres with Proof
  • 6. Sections of Spheres
  • 7. Spherical Lines: Great Circles
  • 8. Soddy Spheres
  • 9. sphere
  • 10. rings on sphere
  • 11. Parametric Sphere
  • 12. Riemann Sphere
  • 13. Bicylinder and Sphere: Cross Sections
  • 14. spheres
  • 15. Sphere
  • 16. A Regular Quadrilateral on a Sphere
  • 17. Sphere througth four points
  • 18. Samsun-Turkey Sphere
  • 19. Arc on sphere
  • 20. sphere
  • 21. Sphere w/ center through point
  • 22. Spheres
  • 23. curve of two spheres Intersection
  • 24. Spheres
  • 25. sphere
  • 26. 3D translated sphere
  • 27. Celestial Sphere
  • 28. Butterbaugh 3D sphere
  • 29. Möbius transformation
  • 30. curve of intersection of a sphere and hyperbolic paraboloid
  • 31. Surface Area Sphere
  • 32. The Volume of the Sphere
  • 33. portion of sphere
  • 34. Mae's Sphere Surface Area
  • 35. Sphere Surface Area
  • 36. Surface Area Sphere
  • 37. Elliptic Coaxial Family of Spheres
  • 38. Sphere
  • 39. Double Box-lid (Mouhefanggai) 牟合方蓋面面觀
  • 40. Surface area right spheres
  • 41. Stereographic projection
  • 42. Locations on a Sphere
  • 43. Shortest Arc
  • 44. Surface Area of Spheres
  • 45. Tracing Double Box-lid (Mouhefanggai) 追蹤蓋合方蓋
  • 46. Define A Sphere with 4 Points Minimal Effort
  • 47. The 24-Point Sphere
  • 48. Sphere
  • 49. Orange slice-2
  • 50. Platonic Solids and spheres
  • 51. triangle on a sphere (sum of angles)

1.

Volume of Spheres with Proof

The figure shows a hemisphere of radius [i]r[/i] and a cylinder of base radius and height [i]r[/i] with an inverted cone of the same height and base radius removed. Drag the red point to see the cross-sections of the two solids at a height [i]h[/i].[br](a) Express [i]x[/i] and [i]y[/i] in terms of [i]r[/i] and [i]h[/i].[br](b) Are the cross-sections equal in area?[br](c) Hence show that the volume of the sphere of radius [i]r[/i] is 4/3 π [i]r[/i]³.
– Anthony OR 柯志明

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