A regular tetrahedron has four equilateral triangles as faces. A regular square pyramid has four equilateral triangles and a square as faces. The faces of the tetrahedron are congruent to the triangular faces of the square pyramid.

A new polyhedron is created by gluing the tetrahedron and the square pyramid together at a triangular face so that the vertices of the triangles coincide. How many faces does this polyhedron have?

At first one might guess 7 faces as 2 faces vanish.
But the polyhedron has only 5 faces as two rhombics are created.
To see this place two pyramids as shown in the figure. (ABCDX and CDEFY)

A D E
X Y
B C F

We know CX=DX=CD=CY=DY. Since CDEFY is a translation of ABCDX, then XY=CD and CDXY is a regular tetrahedron.

The new polyhedron is formed by ABCDXY. Since AD is parallel to XY, ADXY is a single face. Similarily BCXY is a single face. The other three faces of this solid are ABCD, ABX, and CDY. The new polyhedron has five faces total.

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