square based pyramid
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If it is a polyhedron, then four or more faces and four or more corners.
A quadrilateral based pyramid.
A pyramid. It has a square base (4 corners) and four faces converging to a point (the fifth corner). You might be referring to a pentagon which has five sides and five corners.
One possible pair is a cube and a parallelepiped. Vertices and corners are the same thing!
A cone has faces but no corners.