Surface area is a measure of the total area that the surface of a three-dimensional object occupies. It is the sum of all the areas of the faces of the object. The surface area of a shape can be calculated using different formulas depending on the shape of the object. For example, the surface area of a circle is the total area covered by the boundary of a circle, and it can be calculated using the formula A = πr^2, where r is the radius of the circle.

The surface area of a solid object with curved surfaces is more complicated to calculate than the surface area of objects with flat polygonal faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.

The surface area is classified into two categories: lateral surface area or curved surface area, and total surface area. The lateral surface area is calculated to find the area occupied by the curved surfaces, while the total surface area considers all the faces of the 3D shape, including the flat surfaces and the curved surfaces.

In summary, surface area is a measure of the total area that the surface of a three-dimensional object occupies, and it can be calculated using different formulas depending on the shape of the object.