If you are willing to understand what are the faces of a general convex set $C$, you might start to look at what are called the exposed faces of a convex set, which is a notion easy to handle.
Exposed faces are a subclass of faces which are particularly easy to define: $F \subset C$ is an exposed face if it is the intersection of the boundary of $C$ together with some hyperplane being tangent to $C$.
In the case of your disc, you see that any hyperplane will be tangent to the disc at a unique point on the boundary, meaning that each point of the unit circle is a face.
Maybe are you familiar with the notion of face for polyhedra? It happens that for polyhedra the notion of face and exposed face coincide. For example, you can find all the faces of a square by intersecting its boundary with appropriate hyperplane.