---
book: 12
number: 6
id: "XII.6"
kind: "theorem"
uses: ["[[book-12/proposition-5]]", "[[book-5/proposition-18]]", "[[book-5/proposition-22]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.6"
license: "CC-BY-SA-4.0"
---

# XII.6

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

## Proof

Let there be pyramids of the same height of which the polygons *ABCDE*, *FGHKL* are the bases and the points *M*, *N* the vertices; I say that, as the base *ABCDE* is to the base *FGHKL*, so is the pyramid *ABCDEM* to the pyramid *FGHKLN*.

For let *AC*, *AD*, *FH*, *FK* be joined.

Since then *ABCM*, *ACDM* are two pyramids which have triangular bases and equal height, they are to one another as the bases; [[book-12/proposition-5|XII. 5]] therefore, as the base *ABC* is to the base *ACD*, so is the pyramid *ABCM* to the pyramid *ACDM*.

And, componendo, as the base *ABCD* is to the base *ACD*, so is the pyramid *ABCDM* to the pyramid *ACDM*. [[book-5/proposition-18|V. 18]]

But also, as the base *ACD* is to the base *ADE*, so is the pyramid *ACDM* to the pyramid *ADEM*. [[book-12/proposition-5|XII. 5]]

Therefore, ex aequali, as the base *ABCD* is to the base *ADE*, so is the pyramid *ABCDM* to the pyramid *ADEM*. [[book-5/proposition-22|V. 22]]

And again componendo, as the base *ABCDE* is to the base *ADE*, so is the pyramid *ABCDEM* to the pyramid *ADEM*. [[book-5/proposition-18|V. 18]]

Similarly also it can be proved that, as the base *FGHKL* is to the base *FGH*, so is the pyramid *FGHKLN* to the pyramid *FGHN*.

And, since *ADEM*, *FGHN* are two pyramids which have triangular bases and equal height, therefore, as the base *ADE* is to the base *FGH*, so is the pyramid *ADEM* to the pyramid *FGHN*. [[book-12/proposition-5|XII. 5]]

But, as the base *ADE* is to the base *ABCDE*, so was the pyramid *ADEM* to the pyramid *ABCDEM*.

Therefore also, ex aequali, as the base *ABCDE* is to the base *FGH*, so is the pyramid *ABCDEM* to the pyramid *FGHN*. [[book-5/proposition-22|V. 22]]

But further, as the base *FGH* is to the base *FGHKL*, so also was the pyramid *FGHN* to the pyramid *FGHKLN*.

Therefore also, *ex aequali*, as the base *ABCDE* is to the base *FGHKL*, so is the pyramid *ABCDEM* to the pyramid *FGHKLN*. [[book-5/proposition-22|V. 22]] Q. E. D.