---
book: 12
number: 5
id: "XII.5"
kind: "theorem"
uses: ["[[book-12/proposition-3]]", "[[book-10/proposition-1]]", "[[book-12/proposition-4]]", "[[book-5/proposition-11]]", "[[book-5/proposition-16]]", "[[book-12/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.5"
license: "CC-BY-SA-4.0"
---

# XII.5

*Pyramids which are of the same height and have triangular bases are to one another as the bases.*

## Proof

Let there be pyramids of the same height, of which the triangles *ABC*, *DEF* are the bases and the points *G*, *H* the vertices; I say that, as the base *ABC* is to the base *DEF*, so is the pyramid *ABCG* to the pyramid *DEFH*.

For, if the pyramid *ABCG* is not to the pyramid *DEFH* as the base *ABC* is to the base *DEF*, then, as the base *ABC* is to the base *DEF*, so will the pyramid *ABCG* be either to some solid less than the pyramid *DEFH* or to a greater.

Let it, first, be in that ratio to a less solid *W*, and let the pyramid *DEFH* be divided into two pyramids equal to one another and similar to the whole and into two equal prisms; then the two prisms are greater than the half of the whole pyramid. [[book-12/proposition-3|XII. 3]]

Again, let the pyramids arising from the division be similarly divided, and let this be done continually until there are left over from the pyramid *DEFH* some pyramids which are less than the excess by which the pyramid *DEFH* exceeds the solid *W*. [[book-10/proposition-1|X. 1]]

Let such be left, and let them be, for the sake of argument, *DQRS*, *STUH*; therefore the remainders, the prisms in the pyramid *DEFH*, are greater than the solid *W*.

Let the pyramid *ABCG* also be divided similarly, and a similar number of times, with the pyramid *DEFH*; therefore, as the base *ABC* is to the base *DEF*, so are the prisms in the pyramid *ABCG* to the prisms in the pyramid *DEFH*. [[book-12/proposition-4|XII. 4]]

But, as the base *ABC* is to the base *DEF*, so also is the pyramid *ABCG* to the solid *W*; therefore also, as the pyramid *ABCG* is to the solid *W*, so are the prisms in the pyramid *ABCG* to the prisms in the pyramid *DEFH*; [[book-5/proposition-11|V. 11]] therefore, alternately, as the pyramid *ABCG* is to the prisms in it, so is the solid *W* to the prisms in the pyramid *DEFH*. [[book-5/proposition-16|V. 16]]

But the pyramid *ABCG* is greater than the prisms in it; therefore the solid *W* is also greater than the prisms in the pyramid *DEFH*.

But it is also less: which is impossible.

Therefore the prism *ABCG* is not to any solid less than the pyramid *DEFH* as the base *ABC* is to the base *DEF*.

Similarly it can be proved that neither is the pyramid *DEFH* to any solid less than the pyramid *ABCG* as the base *DEF* is to the base *ABC*.

I say next that neither is the pyramid *ABCG* to any solid greater than the pyramid *DEFH* as the base *ABC* is to the base *DEF*.

For, if possible, let it be in that ratio to a greater solid *W*; therefore, inversely, as the base *DEF* is to the base *ABC*, so is the solid *W* to the pyramid *ABCG*.

But, as the solid *W* is to the solid *ABCG*, so is the pyramid *DEFH* to some solid less than the pyramid *ABCG*, as was before proved; [[book-12/proposition-2|XII. 2, Lemma]] therefore also, as the base *DEF* is to the base *ABC*, so is the pyramid *DEFH* to some solid less than the pyramid *ABCG*: [[book-5/proposition-11|V. 11]] which was proved absurd.

Therefore the pyramid *ABCG* is not to any solid greater than the pyramid *DEFH* as the base *ABC* is to the base *DEF*.

But it was proved that neither is it in that ratio to a less solid.

Therefore, as the base *ABC* is to the base *DEF*, so is the pyramid *ABCG* to the pyramid *DEFH*. Q. E. D.