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If a cylinder has two edges then a cone by that definition has one edge and one vertice.
solid with 1 edge is a cone .
slant edge is a height of a cone
The geometrical object known as a cone has no planar faces, two curved surfaces, no edges and one vertex. A "real" object often called a cone is a portion of a half cone. That has one base, a curved surface, and one vertex.
A cone has two faces and one edge. The question does not specify vertical what so it is not possible to answer that part of the question.