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). For a solid with one or more designated
bases, the lateral surface
area excludes the bases, while the total surface
area includes them.
The following table gives surface areas 
for some common surfaces. For
the cone, conical frustum, cylinder, pyramid, and pyramidal frustum, the listed area
is lateral; for the other solids, it is total. Here, 
denotes the radius, 
the height, and 
the base perimeter for a pyramid
or the sum of the two base perimeters for a pyramidal frustum. For a spheroid,
and 
are the equatorial and polar semiaxes, respectively, and 
is the eccentricity.
For a torus, 
is the tube radius and 
is the distance from the rotation axis to the center of the
tube. Finally, 
denotes the slant height (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.
| surface |  |
| cone |  |
| conical frustum |  |
| cube |  |
| cylinder |  |
| oblate spheroid |  |
| prolate spheroid |  |
| pyramid |  |
| pyramidal frustum |  |
| sphere |  |
| spherical lune |  |
| torus |  |
| zone |  |
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
 |
(1)
|
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
 |
(2)
|
where 
and 
are tangent
vectors and 
is the cross product. If 
is defined over a region 
, then
 |
(3)
|
where the integral is taken over the entire surface (Kaplan 1992, pp. 245-248).
Writing 
,
, and 
then gives the symmetrical form
 |
(4)
|
where 
is the transformation of 
,
and
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
are coefficients of the first fundamental form (Kaplan 1992, pp. 245-246).
Whose Area Is the Derivative of the Volume." College
Math. J. 34, 350-358, 2003.Fjelstad, P. and Ginchev, I. "Volume,
Surface Area, and the Harmonic Mean." Math. Mag. 76, 126-129,
2003.Kaplan, W.