A face. For example, a cube has 6 faces.
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The flat surface that goes on and on in all directions has no boundaries. This flat surface is called infinite surface. Accordingly, the answer is that the flat surface that goes on and on in all directions is called an infinite surface.
Such a shape probably does not have a specific name.
Each flat surface is a face
If you mean 1 flat surface and 1 vertex, then the only thing I can think of is a cone.
A face.