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It has 5 faces. The "quadrilateral" part of the name describes the shape of the base of the pyramid. A pyramid has 1 face for every side of the base plus an extra one for the base. Thus a quadrilateral pyramid has a quadrilateral or 4 sided shape for the base and so has 4 + 1 = 5 faces.
A pyramid is the simplest shape but there are many other possibilities.
there are no square pyramids. they are not of the same
It isn't a cone at all, technically or otherwise, by definition. A cone has a circular base; a pyramid, a polygonal one. In fact I think it's strictly only a pyramid if it has a quadrilateral base - anything else being a "~hedron" where the "~" part describes the number of faces, such as the Tetrahedron (4 triangular faces).