A polyhedron can have any number of faces 4 or higher.
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A square pyramid is a polyhedron with a square base and triangular sides that meet at a point at the top. This gives this pyramid five faces.
A square pyramid has 5 vertices and 8 edges. A square pyramid has a square for a base and four triangular faces that connect at the top. The top is one vertex. The four corners of the square base are the other four vertices. The square base has four edges. The other four edges extend from the vertex at the top to the vertices of the square base. This pyramid also has 5 faces.
It's an Octohedron, which has 8 triangular faces. Think of one of the great pyramids in Egypt, with a square base and four triangular sides. Take two of these, turn one upside down, and glue them together along the square base. You wind up with 8 triangular faces, and 12 edges!
A TETRAHEDRON, which is commonly referred to a a triangular based pyramid. The jokingly, a sphere. It has two faces, and outside face and an inside face, Ha!!!Ha!!!
A pyramid has a polygonal base with all other sides being triangles. As the Egyptians famously demonstrated, a common pyramid has a square base with four triangles meeting at a vertex, but any polygon - regular or irregular - can be used as the base of such a polyhedron.