No because a quadrilateral is a 4 sided 2 dimensional polygon whereas a pyramid is a 3 dimensional polyhedron
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No because a quadrilateral is a 4 sided 2 dimensional polygon whereas a pyramid is a 3 dimensional polyhedron
There are an infinite number of such polyhedra. Starting with a tetrahedron (triangular base), the next up is the quadrilateral pyramid (quadrilateral base), then the pentagonal pyramid, the hexagonal pyramid and so on.
A quadrilateral based pyramid. A square based pyramid would be a more common form for school maths.
A square based pyramid is the best name. It is a pentahedron, but that could also refer to a pyramid on any quadrilateral base and so the name is far less informative.
It could be a pyramid with a quadrilateral base or a triangular prism. There are, of course, other related shapes that it could be.
It can be a square, a trapezium, a quadrilateral or a triangle - depending on the inclination of the plane which defines the cross section.