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). 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 S 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, r denotes the radius, h the height, and p the base perimeter for a pyramid or the sum of the two base perimeters for a pyramidal frustum. For a spheroid, a and c are the equatorial and polar semiaxes, respectively, and e is the eccentricity. For a torus, a is the tube radius and c is the distance from the rotation axis to the center of the tube. Finally, s 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.
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
| [画像: S=(dV)/(dr) ] |
(1)
|
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
| S=int_S|T_uxT_v|dudv, |
(2)
|
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, Lateral Surface Area, Pappus's Centroid Theorem, Surface, Surface Integral, Surface of Revolution, Surface Parameterization, Total Surface Area, Volume Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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References
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.Referenced on Wolfram|Alpha
Surface AreaCite this as:
Weisstein, Eric W. "Surface Area." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SurfaceArea.html