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?


1 Answer 1


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 $k=...$


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