A certain polyhedron has 6 vertices and 9 edges. Determine the number of faces on this polyhedron.

Descates (in 1639) has given us the relation between the  vertices , faces and edges of any polyhedron  or any plane figure bounded by sraight edges, plane polygonal faces and vertices.It is popularly known as Euler's formula(Euler rediscovered it in 1751) or Euler's theorem and proved by Cahchy  also in 1811 :

The number of vertices+ number of faces = number of edges +2. Or V+F = E+2. Substitute the given values , V=6 and E = 9  and we get: 6+F = 9+2 . So F = 9+2-6  = 5..