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Pyra3m

solid, content and pyramid

PYRA3M), in geometry, a solid, stand ing on a triangular, square, or polygonal basis, and terminating in a point at the top ; or, according to Euclid, it is a solid figure, consisting of several triangles, whose bases are all in the same plane, and have one common vertex.

Hence the superficies of a given pyra mid is easily found by measuring these triangles separately ; for their sum added to the area of the base is the surface of the pyramid required. It is no less easy to find the solid content of a given pyra mid ; for the area of the base being found, let it be multiplied by the third part of the height of the pyramid, or the third part of the base by the height, and the pro duct will give the solid content, as is de monstrated. by Euclid, lib. 12. prop. 7. If the solid content of afrustrum ofa pyramid is required, first let the solid content of the whole pyramid be found ; from which subtract the solid content of the part that is wanting, and the solid content of the frustrum, or broken pyramid, will remain.

Every pyramid is equal to one third of its circumscribing prism, or that has the same base and height ; that is, the solid content of the prism is equal to one third of the prism For supposing the base a square, then does the pyramid consist of an infinite number of such squares, whose sides, or roots, are continually increasing in arithmetical progression, beginning at the vertex or point, its base being the greatest term, and its perpendicular height the number of all the terms : but the last term multiplied into the number of terms will be triple the sum of all the series, equal the solid content of the py ramid.

All pyramids are in a ratio compound ed of their bases and altitudes ; so that if their bases be equal, they are in propor tion to their altitudes ; and vice versa. Equal pyramids reciprocate their bases and altitudes ; that is, the altitude of one is to that of the other, as the base of the one is to that of the other,