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12 13 cavalIer cone carvIng serious dissection in action Cones come in all sorts of shapes and sises Piles of sand, limpet shells, pyramids, church spires, crystal tips and unicorn horns are all examples of cones. Every cone has a vertex and a base which can be any plane figure. Imagining the vertex as a beacon, a point is in the cone if its shadow is in the base. Let us prove that the formula for the volume of a cone is 1 3 . base area . height, implying the volume formul below. A little shadow play opposite, top illustrates that all cones of the same height and base area are cut by any horisontal plane in slices of equal area. Now, Cavalieris principle see previous page tells us that ALL these cones have the same volume. It is therefore enough to calculate the volume of ONE of these cones such as that of the rightangled pyramid opposite, middle. This and the other two pyramids combine into the triangular prism. Since all three pyramids have equal volume, this volume is one third of the volume of the prism. Q.E.D. To cut a cube into six triangular pyramids of equal volume, first slice it with a diagonal plane into two triangular prisms and then these as above into three pyramids each. Or cut it into three identical square pyramids lower, opposite and then each of these into a pyramid P3 and its mirror image. These six pieces made from paper make a nice puzzle.
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