[Math] How many faces does the resulting polyhedron have

Question

Take a regular tetrahedron of edge one.

Also take a square-based pyramid, whose edges are all one (therefore the side faces are equilateral triangles of same size as the faces of the tetrahedron).

Glue a face of the tetrahedron to a triangular face of the pyramid so that their edges match up.

Considering the volume taken up by the two pieces as a single polyhedron, how many faces does it have?

Answer 1

I visualized the solution Nick provided:

(I found it because it had used this arrangement to build boats models with a magnet game)