The flat surface of a diamond is called the table if it is on the 'top' of a cut diamond. Otherwise, the flat surface is called a facet, some with specific names. Read more, below.
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It is called the surface area.
A flat surface generally refers to a horizontal surface and is referred to as the plane.
A plane surface.
The flat surface of a solid figure is called a face. For example, a rectangular prism has 6 flat surfaces and therefore 6 faces.
The surface of a crystal is called a flat face or a facet.