In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights.

Let there be equal pyramids with triangular bases *ABC* and *DEF* and vertices *G* and *H.*

I say that in the pyramids *ABCG* and *DEFH* the bases are reciprocally proportional to the heights, that is the base *ABC* is to the base *DEF* as the height of the pyramid *DEFH* is to the height of the pyramid *ABCG.*

Complete the parallelepipedal solids *BGML* and *EHQP.*

Now, since the pyramid *ABCG* equals 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* equals the solid *EHQP.*

But in equal parallelepipedal solids the bases are reciprocally proportional to the heights, therefore the base *BM* is to the base *EQ* as the height of the solid *EHQP* is to the height of the solid *BGML.*

But the base *BM* is to *EQ* as the triangle *ABC* is to the triangle *DEF.* Therefore the triangle *ABC* is to the triangle *DEF* as the height of the solid *EHQP* is to the height of the solid *BGML.*

But the height of the solid *EHQP* is identical with the height of the pyramid *DEFH,* and the height of the solid *BGML* is identical with the height of the pyramid *ABCG,* therefore the base *ABC* is to the base *DEF* as the height of the pyramid *DEFH* is to the height of the pyramid *ABCG.*

Therefore in the pyramids *ABCG* and *DEFH* the bases are reciprocally proportional to the heights.

Next, in the pyramids *ABCG* and *DEFH* let the bases be reciprocally proportional to the heights, that is, as the base *ABC* is to the base *DEF,* so let the height of the pyramid *DEFH* be to the height of the pyramid *ABCG.*

I say that the pyramid *ABCG* equals the pyramid *DEFH.*

With the same construction, since the base *ABC* is to the base *DEF* as the height of the pyramid *DEFH* is to the height of the pyramid *ABCG,* while the base *ABC* is to the base *DEF* as the parallelogram *BM* is to the parallelogram *EQ,* therefore the parallelogram *BM* is to the parallelogram *EQ* as the height of the pyramid *DEFH* is to the height of the pyramid *ABCG.*

But the height of the pyramid *DEFH* is identical with the height of the parallelepiped *EHQP,* and the height of the pyramid *ABCG* is identical with the height of the parallelepiped *BGML,* therefore the base *BM* is to the base *EQ* as the height of the parallelepiped *EHQP* is to the height of the parallelepiped *BGML.*

But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal, therefore the parallelepipedal solid *BGML* equals the parallelepipedal solid *EHQP.*

And the pyramid *ABCG* is a sixth part of *BGML,* and the pyramid *DEFH* a sixth part of the parallelepiped *EHQP,* therefore the pyramid *ABCG* equals the pyramid *DEFH.*

Therefore, *in equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights.*

Q.E.D.

When a similar situation appears later where cones are one-third of cylinders, Euclid simply says the same holds for cones, too, with no details whatsoever.

This proposition completes the theory of volumes for pyramids. The next few propositions treat the theory of volumes of cones and cylinders.