Surface Area

Surface area is the Area of a given surface. Roughly speaking, it is the ``amount'' of a surface, and has units of distance squares. It is commonly denoted $S$ for a surface in 3-D, or $A$ for a region of the plane (in which case it is simply called ``the'' Area).

If the surface is Parameterized using $u$ and $v$, then

$$S = \int_S \vert{\bf T}_u\times {\bf T}_v\vert\,du \,dv,$$ (1)

where ${\bf T}_u$ and $\hat{\bf T}_v$ are tangent vectors and ${\bf a}\times{\bf b}$ is the Cross Product.

The surface area given by rotating the curve $y = f(x)$ from $x=a$ to $x=b$ about the $x$-axis is

$$S = \int_b^a 2\pi f(x)\sqrt{1+[f'(x)]^2}\, dx.$$ (2)

If $z = f(x,y)$ is defined over a region $R$, then

$$S = \int\!\!\!\int _R \sqrt{\left({\partial z\over\partial x}\right)^2 +\left({\partial z\over\partial y}\right)^2+1}\,\,dA,$$ (3)

where the integral is taken over the entire surface.

The following tables gives surface areas for some common Surfaces. In the first table, $S$ denotes the lateral surface, and in the second, $T$ denotes the total surface. In both tables, $r$ denotes the Radius, $h$ the height, $p$ the base Perimeter, and $s$ the Slant Height (Beyer 1987).

Surface	$S$
Cone	$\pi r\sqrt{r^2+h^2}$
Conical Frustum	$\pi(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}$
Cube	$6a^2$
Cylinder	$2\pi rh$
Lune (Solid)	$2r^2\theta$
Oblate Spheroid	$2\pi a^2+{\pi b^2\over e}\ln\left({1+e\over 1-e}\right)$
Prolate Spheroid	$2\pi b^2+{2\pi ab\over e}\sin^{-1} e$
Pyramid	${\textstyle{1\over 2}}ps$
Pyramidal Frustum	${\textstyle{1\over 2}}ps$
Sphere	$4\pi r^2$
Torus	$4\pi^2 Rr$
Zone	$2\pi rh$

Surface	$S$
Cone	$\pi r(r+\sqrt{r^2+h^2}\,)$
Conical Frustum	$\pi[{R_1}^2+{R_2}^2+(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}\,]$
Cylinder	$2\pi r(r+h)$

Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution of $y=1/x$ around the x-Axis for $x\geq 1$ is called Gabriel's Horn, and has Finite Volume but Infinite surface Area.

See also Area, Surface Integral, Surface of Revolution, Volume

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127-132, 1987.