A pentagonal cross-section prism would fit the given description
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A pentagonal prism has 7 faces, 15 edges and 10 vertices
It has 10 vertices, 10 edges, and 0 faces.
A pentagonal pyramid has 6 faces, 6 vertices and 10 edges.
7 faces. 10 vertices. 15 edges.
Well, honey, that sounds like a hexagonal prism to me. It's got those six faces, six vertices, and ten edges, making it the life of the 3D shape party. So, go ahead and strut your stuff with that hexagonal prism knowledge!