There is one face on the base and 4 round the edge (from each bottom edge to the top vertex) giving 5 faces in all.
There are 4 edges round the square base and one edge from each of the 4 vertices of the base to the top vertex, giving 8 edges in all.
There are 4 vertices round the square base plus another one at the top, giving 5 vertices in all.
As a check, this obeys Eulers formula: Faces + Vertices = Edges + 2
5 + 5 = 10 = 8 + 2
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5 faces 5 vertices 8 edges .....do the math ;)
A rectangular pyramid has 5 faces, 8 edges and 5 vertices. And it is not vertices's or vertices's's.
A triangular base pyramid has 4 faces, 6 edges and 4 vertices
5 faces, 8 edges and 5 vertices.
A square pyramid has 5 faces, 8 edges and 5 vertices.