A face is a flat surface. The only solid with no flat surfaces is a sphere.
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a sphere * * * * * Or an ellipsoid, or an irregular but smooth "lump".
A contradiction. A solid is a 3-dimensional object, a face is 2-dimensional. Hence you cannot have a solid face.
Its cross-section area
The solid you are describing is called a cylinder. It has two circular faces, one curved face, and no vertices.
A plane.