Pyramid + Tetrahedron = ??

Take two square pyramids and place them on their square bases adjacent to each other. Connect the tips of these two square pyramids and we get a regular tetrahedron between these two shapes. Taking away one square pyramid and counting the number of sides, we get 5.

The pyramid has $5$ faces and the tetrahedron has $4$ faces, but one face of the pyramid gets covered by a face of the tetrahedron, so that makes in principle $5+4-2=7$ faces for the new polyhedron. Though, the problem is not that easy! It turns out that two of the faces of the tetrahedron extend two faces of the pyramid, that is, they make a 180º angle with them. This follows from the fact that the angle $\alpha$ that two faces of the tetrahedron make and the angle $\beta$ that two triangular faces of the pyramid make satisfy $\alpha+\beta=180º$ .

Therefore the solution is $7-2=5$ faces. In fact, the new polyhedron is a prism with base a triangle!