Surface area is the area of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to cover it), and has units of distance squared. Surface area is commonly denoted for a surface in three dimensions, or for a region of the plane (in which case it is simply called "the" area).
The following tables gives lateral surface areas for some common surfaces. Here, denotes the radius, the height, the ellipticity of a spheroid, the base perimeter, the slant height, the tube radius of a torus, and the radius from the rotation axis of the torus to the center of the tube (Beyer 1987). Note that many of these surfaces are surfaces of revolution, for which Pappus's centroid theorem can often be used to easily compute the surface area.
Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution of around the x-axis for is called Gabriel's horn, and has finite volume but infinite surface area.
For many symmetrical solids, the interesting relationship
holds between the surface area , volume , and inradius . This relationship can be generalized for an arbitrary convex polytope by defining the harmonic parameter in place of the inradius (Fjelstad and Ginchev 2003).
If the surface is parameterized using and , then
where and are tangent vectors and is the cross product. If is defined over a region , then
where the integral is taken over the entire surface (Kaplan 1992, pp. 245-248).
Writing , , and then gives the symmetrical form
where is the transformation of , and
are coefficients of the first fundamental form (Kaplan 1992, pp. 245-246).