Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides

Theorem

In the words of Euclid:

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

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

Porism

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 $ABCG$ be a tetrahedron whose base is $\triangle ABC$ and whose apex is $G$.

Let $DEFH$ be a tetrahedron whose base is $\triangle DEF$ and whose apex is $H$.

Let $ABCG$ and $DEFH$ be similar and similarly situated.

It is to be demonstrated that the ratio of $ABCG$ to $DEFH$ is triplicate of the ratio of the sides $BC$ and $EF$.

Let the parallelepipeds $BGML$ and $EHQP$ be completed.

We have that $ABCG$ is similar to $DEFH$.

Therefore:

$\angle ABC = \angle DEF$

$\angle GBC = \angle HEF$

$\angle ABG = \angle DEH$

and:

$AB : DE = BC : EF = BG : EH$

It follows that the parallelogram $BM$ is similar to the parallelogram $EQ$.

For the same reason:

the parallelogram $BN$ is similar to the parallelogram $ER$

and:

the parallelogram $BK$ is similar to the parallelogram $EO$

We also have that:

$BM, BK, BN$ are similar to $EQ, EO, ER$.

But by Proposition $24$ of Book $\text{XI} $: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms:

the three parallelograms $BM, BK, BN$ are equal and similar to their opposites

and

the three parallelograms $EQ, EO, ER$ are equal and similar to their opposites.

Therefore the parallelepipeds $BGML$ and $EHQP$ are contained by equal numbers of similar planes.

Therefore $BGML$ is similar to $EHQP$.

So by Proposition $33$ of Book $\text{XI} $: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides:

$BGML : EQHP = BC^3 : EF^3$

But from:

Proposition $28$ of Book $\text{XI} $: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected

and

Proposition $7$ of Book $\text{XII} $: Prism on Triangular Base divided into Three Equal Tetrahedra

it follows that:

$BGML : EQHP = ABCG : DEFH$

Hence the result.

$\blacksquare$

Historical Note

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

Sources

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