A dodecahedron has $12$ pentagonal faces. How many edges does it have? Each vertex has three pentagons around it. How many vertices are there? Show hint

Each face has $5$ edges, but each edge is shared by two faces.

If we just multiply the number of faces by the number of edges per face, this will overestimate the number of edges by a factor of $2$ because each edge will be counted twice. Therefore, we need to divide by $2$ at the end: \[\frac{12\times5}{2} = 30\] so there are $30$ edges in a dodecahedron.

Similarly, each face has $5$ vertices, but they are shared by three faces, so the total number of vertices is: \[\frac{12\times5}{3} = 20.\]