Pyramids of the same height with polygonal bases are to one another as their bases.

Let there be pyramids of the same height with the polygonal bases *ABCDE* and *FGHKL* and vertices *M* and *N.*

I say that the base *ABCDE* is to the base *FGHKL* as the pyramid *ABCDEM* is to the pyramid *FGHKLN.*

Join *AC, AD, FH,* and *FK.*

Since then *ABCM* and *ACDM* are two pyramids with triangular bases and equal height, therefore they are to one another as their bases. Therefore the base *ABC* is to the base *ACD* as the pyramid *ABCM* is to the pyramid *ACDM.* And, taken together, the base *ABCD* is to the base *ACD* as the pyramid *ABCDM* is to the pyramid *ACDM.*

But the base *ACD* is to the base *ADE* as the pyramid *ACDM* is to the pyramid *ADEM.*

Therefore, *ex aequali,* the base *ABCD* is to the base *ADE* as the pyramid *ABCDM* is to the pyramid *ADEM.*

And again, taken together, the base *ABCDE* is to the base *ADE* as the pyramid *ABCDEM* is to the pyramid *ADEM.* Similarly also it can be proved that the base *FGHKL* is to the base *FGH* as the pyramid *FGHKLN* is to the pyramid *FGHN.*

And, since *ADEM* and *FGHN* are two pyramids with triangular bases and equal heights, therefore the base *ADE* is to the base *FGH* as the pyramid *ADEM* is to the pyramid *FGHN.*

But the base *ADE* is to the base *ABCDE* as the pyramid *ADEM* is to the pyramid *ABCDEM.* Therefore, *ex aequali,* the base *ABCDE* is to the base *FGH* as the pyramid *ABCDEM* is to the pyramid *FGHN.*

But further the base *FGH* is to the base *FGHKL* as the pyramid *FGHN* is to the pyramid *FGHKLN.* Therefore also, *ex aequali,* the base *ABCDE* is to the base *FGHKL* as the pyramid *ABCDEM* is to the pyramid *FGHKLN.*

Therefore, *pyramids of the same height with polygonal bases are to one another as their bases.*

Q.E.D.