How many vertices and edges does it have?

Question

How many vertices and edges does a dodecahedron (a regular convex polyhedron with twelve pentagonal faces) have?

Answer 1

:D is right.
Here's how you can prove it:
There are 5 edges per face.
12 faces x 5 edges/face = 60 edges.
BUT ............. we've counted each edge two times. (There are two faces that touch each edge, and we counted the edge for each of the two faces.) So we have to divide by 2, to get the final answer of 30 edges.

To count vertices:
12 faces x 5 vertices/face = 60 edges.
But 3 faces meet at each vertex, so we've counted each vertex 3 times. So we have to divide by 3, to get a final answer of 20 vertices.

You might be interested to know that the icosahedron (with 20 triangular faces) has 20 faces, 30 edges, and 12 vertices (compared to 12 faces, 30 edges, and 20 vertices for the dodecahedron). So you can build a dodecahedron with an icosahedron inside so that each vertex of the icosahedron touches the middle of a face of the dodecahedron. Or you can put a dodecahedron inside an icosahedron, with each vertex of the dodecahedron touching the center of a face of the icosahedron.

Answer 2

It has twenty vertices and thirty edges.