# A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the same number of edges. What is this common number?

A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the same number of edges. What is this common number?

If I am not mistaken , "this common number" is the number of edges which cover one face.

So to find this i tried following: $$n =$$ # of verices $$e =$$ # of edges $$r =$$ # of regions or faces

$$n = 20$$, $$r = 12$$

We know that polyhydron must be connected plane graph (No intersections of edges and faces). If we apply Euler's Polyhedral Theorem: $$n-e+r=2$$ we get that $$e=30$$. We know that sum of degrees is $$2 e$$, so $$\sum(d(v))=60$$ . Therefore the degree of each vertex is 3. Now i am stuck! What should I do now?

Let $$k$$ be the number of edges per face; then the total number of edges is $$e=\frac{kr}{2}$$ because every edge is adjacent to two faces. We know $$r=12$$ and $$e=30$$, so $$k=...$$