Cavalieri’s principle tells us that a square pyramid has the same volume as a triangular pyramid with a base the same area and equal heights.

Also, it does not matter in which position the apex of the triangular pyramid is.

triangular prism divided into 3 pyramids

A triangular prism ABCDEF is divided into three pyramids P’(ABCD), P’’(CDEF) and P’’’(BEDC).

First, we show that P’ and P’’ have the same volume.

$A_{ABC}$ = $A_{DEF}$ as they are the bases of the prism.

Also,

$\overline{AD}$ is the same length as $\overline{CF}$ and the two pyramids have the same height. By Cavalieri’s principle the pyramids P’ and P’’ have the same volume.

Next, we show that P’’ and P’’’ have the same volume.

Here, $A_{BCE}$ = $A_{DFE}$

as they diagonal divides a parallelogram in half.

And, they share the same height $\overline{DE}$.

Again, according to Cavalieri’s principle the volumes of P’’ and P’’’ are equal.

These three pyramids add up to make the prism which has a volume of

$$V_{\text{prism}} = \cdot A_{\text{G}} \cdot h$$

So, it follows that for the pyramids

$$V_{\text{pyramid}} = \frac{1}{3} \cdot A_{\text{G}} \cdot h$$