Book 12
8
But the pyramid which has a triangular base is to the pyramid which has a triangular base in the triplicate ratio of the corresponding sides; therefore also the pyramid which has a polygonal base has to the pyramid which has a similar base the ratio triplicate of that which the side has to the side.
PROPOSITION 9.
9
In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the heights are equal.
9
For let there be equal pyramids which have the triangular bases ABC, DEF and vertices the points G, H; I say that in the pyramids ABCG, DEFH the bases are reciprocally proportional to the heights, that is, as the base ABC is to the base DEF, so is the height of the pyramid DEFH to the height of the pyramid ABCG.
9
For let the parallelepipedal solids BGML, EHQP be completed.
9
Now, since the pyramid ABCG is equal to the pyramid DEFH, and the solid BGML is six times the pyramid ABCG, and the solid EHQP six times the pyramid DEFH, therefore the solid BGML is equal to the solid EHQP.
9
But in equal parallelepipedal solids the bases are reciprocally proportional to the heights; [XI. 34] therefore, as the base BM is to the base EQ, so is the height of the solid EHQP to the height of the solid BGML.
9
But, as the base BM is to EQ, so is the triangle ABC to the triangle DEF. [I. 34]
9
Therefore also, as the triangle ABC is to the triangle DEF, so is the height of the solid EHQP to the height of the solid BGML. [V. 11]
9
But the height of the solid EHQP is the same with the height of the pyramid DEFH, and the height of the solid BGML is the same with the height of the pyramid ABCG, therefore, as the base ABC is to the base DEF, so is the height of the pyramid DEFH to the height of the pyramid ABCG.
9
Therefore in the pyramids ABCG, DEFH the bases are reciprocally proportional to the heights.
