Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length

Arc length

Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

 of a curve. For polyhedra (objects with flat polygonal faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

) the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, are assigned surface area using their representation as parametric surface

Parametric surface

A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...

s.