Given that P is a square pyramid whose base consists of the four vertices $(0,0,0)$, $(3,0,0)$, $(3,3,0)$, and $(0,3,0)$, and whose apex is the point $(1,1,3)$.
Then let Q be a square pyramid whose base is the same as the base of P, but whose apex is the point $(2,2,3)$.
Find the volume of the intersection of the interiors of P and Q.
Let's call $A=(0,0,0)$, $B=(3,0,0)$, $C=(3,3,0)$, $D=(0,3,0)$ and $V_1=(1,1,3)$, $V_2=(2,2,3)$
Consider the planes $V_2AB$ and $V_2AD$. These planes define the angle of the intersection volume of the two pyramids regarding the sides $AB$ and $AD$, given that $V_2$ is further away from these sides than $V_1$ (hence lower angle).
The same way, $V_1BC$ and $V_1CD$ define the two other angles.
But the intersection of each plane give a line, which will be an edge of the intersection volume. There are four such edges, and the intersection point of these four edges is the apex of the intersection volume, which is also a pyramid.
The apex of the pyramid has the following coordinates: $V_3=(1.5,1.5,2.25)$