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Q: a polyhedron with 12 vertices and 30 edges has how many faces?
A: 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!! [smile] ]
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