# Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides/Porism

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## Porism to Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides

In the words of Euclid:

*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.*

(*The Elements*: Book $\text{XII}$: Proposition $8$ : Porism)

## Proof

Let them be divided into the tetrahedra, by virtue of the fact that the polygons forming their bases can be divided into triangles.

It follows from Proposition $20$ of Book $\text{VI} $: Similar Polygons are composed of Similar Triangles:

- the triangles forming the bases of these tetrahedra are similar.

So from Proposition $12$ of Book $\text{V} $: Sum of Components of Equal Ratios:

- the ratio of one tetrahedron in the one complete pyramid to the corresponding tetrahedron in the other complete pyramid, so is the ratio of all the tetrahedra together in the one complete pyramid to all the tetrahedra together in the other complete pyramid.

- the ratio of one tetrahedron to another tetrahedron is in triplicate ratio to their edges.

Hence the result.

$\blacksquare$

## Historical Note

This proof is Proposition $8$ of Book $\text{XII}$ of Euclid's *The Elements*.

There is reason to doubt that this porism is genuine.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XII}$. Propositions