A cone has one vertex or apex.
A cone is the three dimensional shape produced by the set of possible lines that go through the vertex and also follow the boundary of a particular two dimensional shape, usually a circle, but it can be any shape.
See the Related Links below for the formal definition of a cone and a conical surface, which includes the fact that the vertex can also be called the apex.
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A "vertical" is not a mathematical terms. If you mean vertices, then there's only one.
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.
All triangles have three altitudes.
There is one face in a cone. It is at the bottom.
2 sides, the base and the cone.
1, and it's the base of the cone.