---
book: 12
number: 9
id: "XII.9"
kind: "theorem"
uses: ["[[book-11/proposition-34]]", "[[book-1/proposition-34]]", "[[book-5/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.9"
license: "CC-BY-SA-4.0"
---

# XII.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.*

## Proof

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*.

For let the parallelepipedal solids *BGML*, *EHQP* be completed.

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*.

But in equal parallelepipedal solids the bases are reciprocally proportional to the heights; [[book-11/proposition-34|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*.

But, as the base *BM* is to *EQ*, so is the triangle *ABC* to the triangle *DEF*. [[book-1/proposition-34|I. 34]]

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*. [[book-5/proposition-11|V. 11]]

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*.

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

Next, in the pyramids *ABCG*, *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* is equal to the pyramid *DEFH*.

For, with the same construction, since, as the base *ABC* is to the base *DEF*, so is the height of the pyramid *DEFH* to the height of the pyramid *ABCG*, while, as the base *ABC* is to the base *DEF*, so is the parallelogram *BM* to the parallelogram *EQ*, therefore also, as the parallelogram *BM* is to the parallelogram *EQ*, so is the height of the pyramid *DEFH* to the height of the pyramid *ABCG*. [[book-5/proposition-11|V. 11]]

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

But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal; [[book-11/proposition-34|XI. 34]] therefore the parallelepipedal solid *BGML* is equal to 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* is equal to the pyramid *DEFH*.

Therefore etc Q. E. D.