dodecahedron proof on number of vertices, edges, and faces

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Find, with a proof, the number of vertices, edges, and faces of a dodecahedron.

Its is clear that their are $20$ vertices, $30$ edges, and $12$ faces. I am not sure how to prove this though.

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2 Answers

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For vertices, there are $12$ faces times $5$ vertices per face but since each face is connected to $3$ vertices it is counted three times. Therefore, $V = 12\times 5 \div 3 = 20$.

For edges, there are $12$ faces times $5$ edges per face but since each edge joins $2$ faces it is counted twice. Therefore, $E = 12\times 5 \div 2 = 30$.

Do you see the pattern? Try the same exercise with a tetrahedron and an octahedron to see if you get it.

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Well, depends on what you are allowed to assume. This might be helpful.

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