Volume of pyramid intersection

Question

Suppose that there are two square pyramids on the $xyz$-plane.

Both have base coordinates of $(0,0,0)$, $(30,0,0)$, $(0,30,0)$, and $(30,30,0)$.

One pyramid has its apex at $(10,10,30)$, while the other has its apex at $(20,20,30)$.

What is the volume of their intersection?after change in coordinates

Answer 1

It's just the volume of the first one, because this is entirely below the second.

This is "obvious" if you visualise climbing the "ridgeline" from the origin to the second apex: this ridgeline passes through the first apex, because that is in the same direction from the origin but at half the distance. After you pass the first apex, the second pyramid keeps increasing but the first starts decreasing. So the faces of the two pyramids which lie along the $x$ axis coincide, as do those which lie along the $y$ axis; but the other faces of the first pyramid lie below the corresponding faces of the second.

If this is difficult to visualise you can confirm it algebraically, though it is a fair bit of work. To give one example: consider $(x,y)$ in the triangle with vertices at $(30,0)$, $(20,20)$ and $(10,10)$. One side of this triangle is the line with equation $x+2y=30$, and in the region we have $x+2y\ge30$.

The face of the first pyramid which lies above the region is part of the plane containing $(30,0,0)$, $(30,30,0)$ and $(10,10,10)$: it has equation $$x+2z=30\ .$$ The face of the second pyramid which lies above the region is part of the plane containing $(0,0,0)$, $(30,0,0)$ and $(20,20,20)$, with equation $$y-z=0\ .$$ Therefore we have $$x+2y\ge30\quad\Rightarrow\quad \frac{30-x}{2}\le y\quad\Rightarrow\quad z_1\le z_2\ .$$

Answer 2

It is impossible for the volume of the intersection to be the volume of either pyramid because neither pyramid completely lies within the other. The easiest way to think about this is: Both pyramids have the same height as given in the question (both have 30), but different apexes. Therefore, neither pyramid lies completely within the other.

For one square pyramid to fit within another, one apex must be less than that of the other (among many other requirement), OR , it can be the same height BUT, if it has the same height then it must also share the coordinates with that of the other.