Question

A polyhedron has 12 faces and 30 edges. What is the number of vertices of the polyhedron?

Answer 1

HI,

The polyhedral formula states
V + F - E = 2,

where V is the number of polyhedron vertices, E is the number of polyhedron edges, and F is the number of faces. Since the dodecahedron has 12 sides, so F = 12.

Each of the 12 faces has 5 vertices, for a total of 60 vertices if all the sides were laid out separately. But 3 faces meet at each vertex so 60/3 = 20 as the actual number of vertices. So V = 20.

Each of the 12 faces has 5 edges, for a total of 60 edges if all the sides were laid out separately. But 2 faces share each edge so 60/2 = 30 as the actual number of edges. So E = 30.

These numbers work in the polyhedral formula above:

V + F - E = 2
20 + 12 - 30 = 2
32 - 30 = 2
2 = 2

The dodecahedron has 12 faces, 20 vertices and 30 edges.

I hope that helps!! :-)

Answer 2

This is also called the Euler's Formula, right?

Answer 3

ya

Answer 4

Thanks so much :)
I'm still a bit confused on something; how do you know that the twelve faces has five vertices?