a Plane
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Mathematically, a cone is infinite and so has no flat surface. The popular cone is actually a truncated cone and does have 1 flat surface.
A face is a flat surface of a 3 dimensional figure.
A plane
A plane is a flat, closed figure.a flat surface on which a straight line joining any two points on it would wholly lie:
In geometry, a circle is a two-dimensional object, so it does not have a surface. Only three-dimensional objects have surfaces. A disc is a flat, circular three-dimensional object.