CavalieriTorus

# CavalieriTorus - The comparison is below At a height h from...

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Cavalieri’s Determination of the Volume of a Torus Bonaventura Cavalieri (1598-1647) was a contemporary of Galileo who considered him the greatest geometer since Archimedes. One of his powerful tools has been called Cavalieri’s Principle. It states that if two solids have the same height and at each level have the same cross sectional area, then they both have the same volume. This is represented with a mnemonic diagram below. The volume of a torus can be determined in this way by comparing its cross-sectional areas with that of a cylinder.

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Unformatted text preview: The comparison is below. At a height h from the center of the figures we get the following cross sections. A 1 ( h ) A 2 ( h ) h V 1 V 2 b a torus 2 ! b 2 a cylinder b + a 2 ! h 2 b ! a 2 ! h 2 2 b 2 a 2 ! h 2 So, the area for the torus is A 1 ( h ) = ! b + a 2 " h 2 ( ) 2 " b " a 2 " h 2 ( ) 2 = 4 b a 2 " h 2 . The area for the cylinder is clearly A 2 ( h ) = 4 b a 2 " h 2 . So, A 1 ( h ) = A 2 ( h ) for every h . This implies that the volumes are the same. So, the volume of a torus is given by the following. V = 4 2 ba 2...
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CavalieriTorus - The comparison is below At a height h from...

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