A cone has one face, one edge and no corners. A cone is defined as a hollow or solid object with a circular base that tapers upward to a point.
The circular plane surface of the cone is a flat surface. A cone has one edge at the intersection of the circular plane surface with the curved surface originating from the vertex. Although a cone does taper upwards to a point, that point is not formed by flat surfaces and thus does not constitute a corner.