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Cavalieri’s Principle tells us the volumes of these pyramid equal one another.
2 Initial steps To begin with, we need to define the volume of cube and cuboid. 2.1 Cube
ܽ 3 , where
ܽ is the length of one side.
We postulate the volume of a cube is
2.2 Cuboid
ሺܽ ൈ ܾ ൈ ܿሻ . Then
Suppose three values a, b, c which are the sides of a cuboid
ܽ ܾ ܿ . According to our axiom, the
we form a cube with the side length of
volume of this cube would be ሺܽ ܾ ܿሻ 3 : ሺܽ ܾ ܿሻ 3 ൌ ܽ 3 ܾ 3 ܿ 3 3ܾܽܿሺܽ ܾ ܿሻ ֜ ሺܽ ܾ ܿሻ 3 ܽ 3 ܾ 3 ܿ 3 ൌ ሺܽ ܾሻ 3 ሺܽ ܿሻ 3 ሺܾ ܿሻ 3 6ܾܽܿ ൌ ሺܽ ܾሻ3 ሺܽ ܿሻ3 ሺܾ ܿሻ3 6ܾܽܿ Then we divide the big cube into 3 cubes which have a 3 , b 3 , c 3 cubes intersecting. Therefore, the remaining pieces are 6 cuboids; the volume of each would be ܽ ൈ ܾ ൈ ܿ . Accordingly, the volume of a cuboid would be : ܪ ݄݁݅݃ ݐ ൈ ܮ ݁݊݃ ݐ ݄ ൈ ܹ݅݀ ݐ ݄ . 3 Implementation In this section we try to work out the volume of some common shapes using the introduced principle. 3.1 Pyramid Let’s define “perfect pyramid”. The base of a perfect pyramid is a rectangle and the height is one of its sides.
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