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?
Best Answer
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 $k=...$