Well it has one edge, one face, one base(the face that the shape sits on), and one vertice.
One edge (boundary) and no vertices.
If a cylinder has two edges then a cone by that definition has one edge and one vertice.
Conventionally, two faces, one edge and one vertex.
A finite cone has one vertex (not vertices!), 1 edge and 2 faces.
There is no such shape - unless there are other faces. If there is one rectangular face then you require at least four other faces - one on each edge of the rectangle - making a minimum of 5 faces.
A cone perhaps
A cone would fit the given description
It has one spherical face and no vertices or edges