Q: Does a polyhedron have 10 faces 28 edges and 18 vertices?

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A pentagonal pyramid.

Not any normal polyhedron since the numbers are contary to the Euler characteristic.

There can be no such polyhedron since the given numbers are not consistent with the Euler characteristic.

A pentagonal pyramid has 6 faces, 6 vertices and 10 edges.

A pentagonal prism has 7 faces, 10 vertices and 15 edges.

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For all polyhedra: vertices + faces = edges + 2 The given fact is: edges = vertices + 10 → vertices + faces = vertices + 10 + 2 → faces = 12

Such a polyhedron cannot exist. According to the Euler characteristics, V + F - E = 2, where V = vertices, F = faces, E = edges. This would require that the polyhedron had only two faces.

A pentagonal pyramid.

No. The numbers are not consistent with the requirements of the Euler characteristic.

Faces: 10 Vertices: 16 Edges: 24

Not any normal polyhedron since the numbers are contary to the Euler characteristic.

There can be no such polyhedron since the given numbers are not consistent with the Euler characteristic.

A pentagonal pyramid has 6 faces, 6 vertices and 10 edges.

A pentagonal pyramid has 6 faces, 6 vertices and 10 edges.

A pentagonal prism has 7 faces, 10 vertices and 15 edges.

A decahedron is a polyhedron with 10 faces. There are several versions of a decahedron, but none of these are regular. By definition, they all have 10 faces. There is the octagonal prism - with 24 edges and 16 vertices, the square anti-prism, with 16 edges and 8 vertices, the square cupola, with 20 edges and 12 vertices, the pentagonal bi-pyramid, with 15 edges and 7 vertices and the augmented pentagonal prism, with 19 edges and 11 vertices. See, for example, http://en.wikipedia.org/wiki/Decahedron

A pentagonal based pyramid has 6 faces, ten edges and 6 vertices.