Sizes of Pyramids of Same Height with Polygonal Bases are as Bases

Theorem

In the words of Euclid:

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

Proof

Let there be two pyramids of the same height whose bases are the polygons $ABCDE$ and $FGHKL$ and whose apices are $M$ and $N$.

It is to be demonstrated that the ratio of $ABCDE$ to $FGHKL$ equals the ratio of pyramid $ABCDEM$ to pyramid $FGHKLN$.

Let $AC, AD, FH, FK$ be joined.

We have that $ABCM$ and $ACDM$ are tetrahedra of the same height.

the ratio of $ABCM$ to $ACDM$ equals the ratio of $\triangle ABC$ to $\triangle ACD$.
$ABCD : \triangle ACD = ABCDM : ACDM$
the ratio of $ACDM$ to $ADEM$ equals the ratio of $\triangle ACD$ to $\triangle ADE$.
$ABCD : \triangle ADE = ABCDM : ADEM$
$ABCDE : \triangle ADE = ABCDEM : ADEM$

Similarly it can be shown that:

$FGHKL : \triangle FGH = FGHKLN : FGHN$

We have that $ADEM$ and $FGHN$ are two tetrahedra of the same height.

the ratio of $ADEM$ to $FGHN$ equals the ratio of $\triangle ADE$ to $\triangle FGH$.

But we have:

$\triangle ADE : ABCDE = ADEM : BCDEM$
$ABCDE : \triangle FGH = ABCDEM : FGHN$

We also have:

$\triangle FGH : FGHKL = FGHN : FGHKLN$
$ABCDE : FGHKL = ABCDEM : FGHKLN$

$\blacksquare$

Historical Note

This theorem is Proposition $6$ of Book $\text{XII}$ of Euclid's The Elements.