I am assuming that like in sphere, there is one face only. In the same way, a Cone has 2 faces.

Base. which is circular face.Curved surface as shown in the image below.

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\$egingroup\$ The first link says \$2\$ faces too (read again). The third link says \$1\$ \$flat\$ face. \$endgroup\$
It depends whether an open cone or close cone. An open cone doesn"t include the circular base while a close cone includes.

By the way, a close cylinder should include two circular bases and one curved surface. You may have a semi-open cylinder (e.g. cylindrical pencil stand) which has one base. The core of a toilet paper roll has two ends open, that is one curved surface only.

P.S.:In calculus texts, a cone is approached by a polygonal pyramid. So a curved surface has infinitely many "faces" (that means tiny planes).

In differential geometry or topology, sphere and torus have inner and outer surfaces while Klein bottle is one-sided.

A cone has a flat circular face as its base, and a curved surface seen as its curved face connecting the base boundary to a single point apex... There are two surfaces and an edge connecting them.

The curved cone surface is the sum of an infinite number of flat triangular faces.

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