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

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Theorem

In the words of Euclid:

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

(The Elements: Book $\text{XII}$: Proposition $6$)


Proof

Euclid-XII-6.png

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.

So by Proposition $5$ of Book $\text{XII} $: Sizes of Tetrahedra of Same Height are as Bases:

the ratio of $ABCM$ to $ACDM$ equals the ratio of $\triangle ABC$ to $\triangle ACD$.

From Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded:

$ABCD : \triangle ACD = ABCDM : ACDM$

But by Proposition $5$ of Book $\text{XII} $: Sizes of Tetrahedra of Same Height are as Bases:

the ratio of $ACDM$ to $ADEM$ equals the ratio of $\triangle ACD$ to $\triangle ADE$.

Therefore from Proposition $22$ of Book $\text{V} $: Equality of Ratios Ex Aequali:

$ABCD : \triangle ADE = ABCDM : ADEM$

Again from Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded:

$ABCDE : \triangle ADE = ABCDEM : ADEM$


Similarly it can be shown that:

$FGHKL : \triangle FGH = FGHKLN : FGHN$

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

Therefore from Proposition $5$ of Book $\text{XII} $: Sizes of Tetrahedra of Same Height are as Bases:

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

But we have:

$\triangle ADE : ABCDE = ADEM : BCDEM$

Therefore by Proposition $22$ of Book $\text{V} $: Equality of Ratios Ex Aequali:

$ABCDE : \triangle FGH = ABCDEM : FGHN$

We also have:

$\triangle FGH : FGHKL = FGHN : FGHKLN$

Further by Proposition $22$ of Book $\text{V} $: Equality of Ratios Ex Aequali:

$ABCDE : FGHKL = ABCDEM : FGHKLN$

$\blacksquare$


Historical Note

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


Sources