Volume

Arthur Lee, Nov 27, 2014

This set of models can be used for discussing about volume of solids.

Table of Contents

  1. pyramid
    1. three tetrahedrons
    2. Volume of Pyramids
    3. equal cross sections
    4. opening a cube
    5. tisect a cube
    6. Trisecting the Cube into 3 Pyramids
    7. Volume of Pyramids - Chinese Proof
  2. sphere
    1. cutting a sphere
    2. section of sphere
    3. section of cone
    4. Volume of Spheres
    5. A sphere in a cylinder.
    6. Kugelvolumen - Cavalieri

1.

pyramid

  • 1. three tetrahedrons
  • 2. Volume of Pyramids
  • 3. equal cross sections
  • 4. opening a cube
  • 5. tisect a cube
  • 6. Trisecting the Cube into 3 Pyramids
  • 7. Volume of Pyramids - Chinese Proof

1.1.

three tetrahedrons

A triangular prism is divided into 3 tetrahedrons (or triangular pyramids). Drag G or H to fit the solids inside the prism. [br]These tetrahedrons are different in shape but their volumes are equal.[br]Drag A, B or C to change the base, or D to change the height.
To compare the volume of any pair of tetrahedrons, find two triangular faces that are congruent, one from each tetrahedron. Then identify their corresponding heights.
– Arthur Lee

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1.5.

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1.6.

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1.7.

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2.

sphere

  • 1. cutting a sphere
  • 2. section of sphere
  • 3. section of cone
  • 4. Volume of Spheres
  • 5. A sphere in a cylinder.
  • 6. Kugelvolumen - Cavalieri

2.1.

cutting a sphere

A sphere, a cone and a cylinder of same radius are cut by the same horizontal plane. The height of the cone and cylinder is equal to their radius. [br]We can compare their cross sections obtain from the height. [br]Drag G to change the height of the cutting plane.
cutting a sphere
Regardless of the height of the cutting plane, the sum of sections from the sphere and cone is the equal to the section of the cylinder. [br]By the Cavalieri's Principle*, the volume of the cylinder should be equal to the sum of volume of the cone and the hemisphere.[br][br]* [url]http://en.wikipedia.org/wiki/Cavalieri's_principle[/url]
– Arthur Lee

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2.6.

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