Eulers formula for graphs doesn't seem to work for a triangle? I'm obviously missing something obvious here. Euler's formula for planar graphs is:
$$v-e+f = 2$$
Consider a triangle. It has three vertices and three edges. Substituting this into the formula above gives $f = 2$. Maybe this is because the region inside the triangle and that outside is counted as two faces. Now, for a Tetrahedron, the number of vertices is four and edges is six. Substituting this above we get $f = 4$. But this is exactly the number of faces a Tetrahedron has. We didn't have to split it into two regions. What am I missing? 

Yes, it's that you count the outside region as a face.
What happens if you try drawing a tetrahedron in the plane? You end up with something like the picture, and one of the four faces becomes the "outside" face.

