A normal cone has a circular face as its base, and a slanting surface which goes from the circumference of this face to the vertex. So it has two faces.
Mathematicians also refer to the infinite equivalent as a cone. This could be considered as a 3-d shape with one face.
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Cone or a pyramid
You're probably thinking of a cone. There is some disagreement as to whether the point of a cone qualifies as a vertex.
cone
cone
A cone.