Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.  This could include Surface area (for 2D-surfaces),  the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.
Area could also mean the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

vCalc's Area folder contains several well-known formulas for converting Area into different Area units.