G.542 parallelepipedal solids height — Heath

XI.32

Parallelepipedal solids which are of the same height are to one another as their bases.

Let < A B >, < C D > be parallelepipedal solids of the same height; I say that the parallelepipedal solids < A B >, < C D > are to one another as their bases, that is, that, as the base < A E > is to the base < C F >, so is the solid < A B > to the solid < C D >.

For let < F H > equal to < A E > be applied to / F G /, [G.74] and on < F H > as base, and with the same height as that of < C D >, let the parallelepipedal solid < G K > be completed.

Then the solid < A B > is equal to the solid < G K >; for they are on equal bases < A E >, < F H > and of the same height. [G.541]

And, since the parallelepipedal solid < C K > is cut by the plane < D G > which is parallel to opposite planes,

therefore, as the base < C F > is to the base < F H >, so is the solid < C D > to the solid < D H >. [G.531]

But the base < F H > is equal to the base < A E >, and the solid < G K > to the solid < A B >;

therefore also, as the base < A E > is to the base < C F >, so is the solid < A B > to the solid < C D >.

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.543, G.544, G.547, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578