Question

A polyhedron has 30 edges and 20 vertices. How many faces does this polyhedron have?

Answer 1

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{edges},\mathrm{E}=30 \\ \mathrm{Number}\mathrm{of}\mathrm{vertices},\mathrm{V}=20 \\ \mathrm{Number}\mathrm{of}\mathrm{faces},\mathrm{F}=? \\ \mathrm{Now},\mathrm{using}\mathrm{Euler}\text{'}\mathrm{s}\mathrm{formula}: \\ \mathrm{V}+\mathrm{F}-\mathrm{E}=2 \\ \Rightarrow 20+\mathrm{F}-30=2 \\ \Rightarrow \mathrm{F}-10=2 \\ \Rightarrow \mathrm{F}=10+2 \\ \Rightarrow \mathrm{F}=12 \end{gathered}$$