Cones and pyramids have only one base, but prisms have multiple bases.
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A cone has two faces: one around (the curved part), and one on the flat base. It has one vertex at the point at the top and one edge where the curved section meets the flat base.
No, prisms and pyramids do not have the same number of vertices. A prism has two identical polygonal bases connected by rectangular faces, so it has 2 more vertices than the number of sides in the base polygon. A pyramid has a polygonal base and triangular faces connecting the base to a single vertex, so it has 1 more vertex than the number of sides in the base polygon.
One difference is square pyramids have 4 triangular faces while rectangular prisms have none!