Surface Area

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 S for a surface in three dimensions, or A for a region of the plane (in which case it is simply called "the" area).

The following tables gives lateral surface areas S for some common surfaces. Here, r denotes the radius, h the height, e the ellipticity of a spheroid, p the base perimeter, s the slant height, a the tube radius of a torus, and c 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 y=1/x around the x-axis for x>=1 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 S, volume V, and inradius r. This relationship can be generalized for an arbitrary convex polytope by defining the harmonic parameter h in place of the inradius r (Fjelstad and Ginchev 2003).

If the surface is parameterized using u and v, then


where T_u and T_v are tangent vectors and axb is the cross product. If z=f(x,y) is defined over a region R, then


where the integral is taken over the entire surface (Kaplan 1992, pp. 245-248).

Writing x=x(u,v), y=y(u,v), and z=z(u,v) then gives the symmetrical form


where R^' is the transformation of R, and


are coefficients of the first fundamental form (Kaplan 1992, pp. 245-246).

See also

Area, Area Element, Fundamental Forms, Harmonic Parameter, Pappus's Centroid Theorem, Surface, Surface Integral, Surface of Revolution, Surface Parameterization, Volume Explore this topic in the MathWorld classroom

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Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127-132, 1987.Dorff, M. and Hall, L. "Solids in R^n 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. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, 1992.

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Surface Area

Cite this as:

Weisstein, Eric W. "Surface Area." From MathWorld--A Wolfram Web Resource.

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