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

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

From :
 * $ABCD : \triangle ACD = ABCDM : ACDM$

But by :
 * the ratio of $ACDM$ to $ADEM$ equals the ratio of $\triangle ACD$ to $\triangle ADE$.

Therefore from :
 * $ABCD : \triangle ADE = ABCDM : ADEM$

Again from :
 * $ABCDE : \triangle ADE = ABCDEM : ADEM$

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 :
 * 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 :
 * $ABCDE : \triangle FGH = ABCDEM : FGHN$

We also have:
 * $\triangle FGH : FGHKL = FGHN : FGHKLN$

Further by :
 * $ABCDE : FGHKL = ABCDEM : FGHKLN$