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?

(In reply to Surely not the intended solution ! by Syzygy)

I was thinking like you there for a minute...

My first instinct was to say "duh, it’s 7." You lose one side of the tetrahedron (so you have 3 faces now) and you lose one side of the pyramid (so you have 4 faces now). So you get 7 faces.

But then I thought "this is perplexus… that was a little too easy." And while I know we have some simple problems on here, I just couldn’t believe I was right.

So I considered the following… what if, after the gluing process, 2 faces of the tetrahedron were actually in the same planes as 2 triangular faces of the pyramid? Then they would "blend" together and you really only have a 5 sided polyhedron (1 square face, 2 diamond faces, and two triangular faces).

I didn’t do the math to find the angles, but that’s what I think the answer is.

Posted by nikki on 2004-10-05 13:10:14