---
book: 12
number: 8
id: "XII.8"
kind: "theorem"
uses: ["[[book-11/proposition-24]]", "[[book-11/proposition-33]]", "[[book-11/proposition-28]]", "[[book-12/proposition-7]]", "[[book-6/proposition-20]]", "[[book-5/proposition-12]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.8"
license: "CC-BY-SA-4.0"
---

# XII.8

*Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.*

## Proof

Let there be similar and similarly situated pyramids of which the triangles *ABC*, *DEF*, are the bases and the points *G*, *H* the vertices; I say that the pyramid *ABCG* has to the pyramid *DEFH* the ratio triplicate of that which *BC* has to *EF*.

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

Now, since the pyramid *ABCG* is similar to the pyramid *DEFH*, therefore the angle *ABC* is equal to the angle *DEF*, the angle *GBC* to the angle *HEF*, and the angle *ABG* to the angle *DEH*; and, as *AB* is to *DE*, so is *BC* to *EF*, and *BG* to *EH*.

And since, as *AB* is to *DE*, so is *BC* to *EF*, and the sides are proportional about equal angles, therefore the parallelogram *BM* is similar to the parallelogram *EQ*.

For the same reason *BN* is also similar to *ER*, and *BK* to *EO*; therefore the three parallelograms *MB*, *BK*, *BN* are similar to the three *EQ*, *EO*, *ER*.

But the three parallelograms *MB*, *BK*, *BN* are equal and similar to their three opposites, and the three *EQ*, *EO*, *ER* are equal and similar to their three opposites. [[book-11/proposition-24|XI. 24]]

Therefore the solids *BGML*, *EHQP* are contained by similar planes equal in multitude.

Therefore the solid *BGML* is similar to the solid *EHQP*.

But similar parallelepipedal solids are in the triplicate ratio of their corresponding sides. [[book-11/proposition-33|XI. 33]]

Therefore the solid *BGML* has to the solid *EHQP* the ratio triplicate of that which the corresponding side *BC* has to the corresponding side *EF*.

But, as the solid *BGML* is to the solid *EHQP*, so is the pyramid *ABCG* to the pyramid *DEFH*, inasmuch as the pyramid is a sixth part of the solid, because the prism which is half of the parallelepipedal solid [[book-11/proposition-28|XI. 28]] is also triple of the pyramid. [[book-12/proposition-7|XII. 7]]

Therefore the pyramid *ABCG* also has to the pyramid *DEFH* the ratio triplicate of that which *BC* has to *EF*. Q. E. D.

Porism. From this it is manifest that similar pyramids which have polygonal bases are also to one another in the triplicate ratio of their corresponding sides.

For, if they are divided into the pyramids contained in them which have triangular bases, by virtue of the fact that the similar polygons forming their bases are also divided into similar triangles equal in multitude and corresponding to the wholes [[book-6/proposition-20|VI. 20]], then, as the one pyramid which has a triangular base in the one complete pyramid is to the one pyramid which has a triangular base in the other complete pyramid, so also will all the pyramids which have triangular bases contained in the one pyramid be to all the pyramids which have triangular bases contained in the other pyramid [[book-5/proposition-12|V. 12]], that is, the pyramid itself which has a polygonal base to the pyramid which has a polygonal base.

But the pyramid which has a triangular base is to the pyramid which has a triangular base in the triplicate ratio of the corresponding sides; therefore also the pyramid which has a polygonal base has to the pyramid which has a similar base the ratio triplicate of that which the side has to the side.