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A ____ has a regular polygon base and a vertex over the center of the base.

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That would be a regular pyramid my friend!

A regular pyramid has a regular polygon base and a vertex over the center of the base.

The shape of a regular pyramid is described as a polyhedron. It is formed by joining a polygonal base with an apex. A pyramid is a conic solid with its top in the form of a sharp cone. There are 3 lateral faces of the pyramid which is formed using one base edge with an apex.