What is the euler characteristic of a polyhedron that has 6 vertices, 12 edges, and 7 faces?

The Euler formula is Vertices-Edges+Faces so lets plug it in.
6-12+7=
1 is the answer

Regardless of the form of the hexahedron, for simply-connected spaces, the Euler Characteristic is 2, giving the formula 2 = V - E + F, where V is the number of vertices, E is the number of edges and F is the number of faces. In this case we have 2 = V - 12 + 6, or 
V = 2 + 12 - 6 = 8.