Everything about Surface Area totally explained
Surface area is the measure of how much exposed
area an object has. It is expressed in square units. If an object has flat
faces, its surface area can be calculated by adding together the areas of its faces. Even objects with smooth surfaces, such as
spheres, have surface area.
Formulas
Sphere:
The surface area of a sphere is the
integral of infinitesimal circular rings of width
The radius of the circular ring is
The
antiderivative needed is the simple linear function
Thus, the sphere surface area amounts to
A
sphere =
Surfaces whose area can't be defined
While areas of many simple surfaces have been known since antiquity, a rigorous mathematical
definition of area requires a lot of care. Various approaches to defining the surface area were developed in the late nineteenth and the early twentieth century by
Henri Lebesgue and
Hermann Minkowski. For a very wide class of geometric surfaces called
piecewise-smooth all these approaches result in the same notion of area. However, if a surface is very irregular, or rough, then it may not be possible to assign any area at all to it. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the theory of
fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in the
geometric measure theory. A specific example of such an extension is the
Minkowski content of a surface.
In chemistry
Surface area is important in
chemical kinetics. Increasing the surface area of a substance generally increases the
rate of a
chemical reaction. For example,
iron in a fine powder will
combust, while in solid blocks it's stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.
In biology
The surface area-to-volume ratio (SA:V) of a
cell imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the
cell membrane to interstitial spaces or to other cells. If you consider the math, you'll see the relation between SA and V much more intuitively: V = 4/3 π r
3; SA = 4 π r
2, where r is the radius of the cell. Do the math and the resulting ratio becomes 3/r. If a cell has a radius of 1 μm, the SA:V ratio is 3. Increase the cell's radius to 10 μm and the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Using the previous simple example, we can see how the surface area falls off steeply with increasing volume.
Further Information
Get more info on 'Surface Area'.
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