# Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms

### Euclid's Formulation

If there are two pyramids with the same height, having trianglular bases, and each of them is divided into two pyramids equal to one another, and similar to the whole, and into two equal prisms then as the base of one pyramid (is) to the base of the other pyramid, so (the sum of) all the prisms in one pyramid will be to (the sum of all) the equal number of prisms in the other pyramid.

• Let there be two pyramids with the same height, having the triangular bases $ABC$ and $DEF$, (with) apexes the points $G$ and $H$ (respectively).
• And let each of them have been divided into two pyramids equal to one another, and similar to the whole, and into two equal prisms [Prop. 12.3].
• I say that as base $ABC$ is to base $DEF$, so (the sum of) all the prisms in pyramid $ABCG$ (is) to (the sum of) all the equal number of prisms in pyramid $DEFH$.

### Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick