Area is a measure of the extension of a surface expressed in units known as surface units. For flat surfaces the concept is more intuitive. Any flat surface with straight sides can triangulate and its area can be calculated as the sum of the areas of these triangles. The term "Area is cccasionally used as a synonym for surface, when there is no confusion between the geometrical concept itself (surface) and the magnitude metric associated to geometric concept (area).
However, to calculate the curved surface area is required to introduce methods of differential geometry.
To define the area of a surface in general (which is a metric concept), it has to be defined a metric tensor on the surface in question: when the surface is in a Euclidean space, the surface naturally inherits a metric structure induced by the Euclidean metric.