A pentagonal pyramid.
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It is a pentagonal based pyramid that has 6 vertices, 6 faces and 10 edges
A shape that has fewer than 8 faces and fewer than 10 edges is a triangular prism. A triangular prism has 5 faces (2 triangular faces and 3 rectangular faces) and 9 edges. It meets the criteria of having fewer than 8 faces and fewer than 10 edges.
There is no simple way. You cannot go by the name - which can refer to the number of faces. You can have different shapes with the same number of faces. The following are all examples of a hexahedron (six faces): triangular dipyramid (9 edges), pentagonal prism (10 edges), rectangular prism (12 edges). One contributor has suggested it might be easier to count the faces and vertices (corners). Add the two numbers and then subtract 2. *This will work if the shape has been simplified as much as it can be. The 2 is only valid if there are no holes in the shape. For example, a pentagonal prism: 6 faces, 6 vertices. Sum = 12. Subtract 2 = 10. A pentagonal pyramid has 10 edges.
A heptagonal prism gas 21 edges.
PENTAGONAL prism