# Surface area

﻿
Surface area

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.

Formula

Sphere

The surface area of a sphere is the integral of infinitesimal circular rings of width $dx$ The radius of the circular ring is $f\left(x\right) = sqrt\left\{r^2-x^2\right\}$. The length of the circular ring is equal to $2picdot f\left(x\right)$ The width of the ring can be determined by using Pythagoras' formula for a rectangular triangle with side lengths $dx$ and $f\text{'}\left(x\right) cdot dx$, which leads to $sqrt\left\{1+f\text{'}\left(x\right)^2\right\},dx$ The infinitesimal surface area of the circular ring thus is equal to $2pi f\left(x\right)cdot sqrt\left\{1+f\text{'}\left(x\right)^2\right\},dx$ The derivative of $f\left(x\right)$ is equal to $f\text{'}\left(x\right) = frac\left\{-x\right\}\left\{sqrt\left\{r^2-x^2$ The surface area of the sphere can be calculated as $int_\left\{-r\right\}^r 2pi f\left(x\right)cdot sqrt\left\{1+f\text{'}\left(x\right)^2\right\},dx$ = $int_\left\{-r\right\}^r 2pi sqrt\left\{r^2-x^2\right\} cdot sqrt\left(1+frac\left\{x^2\right\}\left\{r^2-x^2\right\}\right),dx = int_\left\{-r\right\}^r 2pi sqrt \left\{r^2\right\},dx = 2pi r int_\left\{-r\right\}^r 1,dx$ The antiderivative needed is the simple linear function $x$ Thus, the sphere surface area amounts to Asphere = $2pi r \left[r-\left(-r\right)\right] = 4pi r^2$

=Cylinder=

The surface area of a (circular) cylinder of radius "r" and height "h" is:$2 pi r h + 2 pi r^2$where the second term shows the contributions of the top and bottom of the cylinder.

Cube

Sa = Total Surface Area, L = Side Length, V = Volume, Sf = Surface Area Of a Single Face
$Sa = L^26$
$Sa = Sf6$
$Sa = \left(V^\left\{-3\right\}\right)^26$

Surfaces whose area cannot 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 is 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 r3; SA = 4 π r2, 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.

What limitations does this place on a living cell? For small cells, SA:V ratio allows for relatively good exchange of nutrients and wastes. For larger cells and organisms, SA:V ratio forces the cell or organism to find more efficient ways to exchange nutrients and waste products, e.g. specific conduits that carry blood, hormones, lymph, etc. from deep regions to the surface of an organism.

* [http://www.thinkanddone.com/ge/SurfaceArea.html Compute Surface Area with online utility ]

*Synthetic geometry

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• surface area — surface ,area noun count the total area of a surface or surfaces, especially the outside surfaces of an object: enough paint to cover a surface area of 900 square feet …   Usage of the words and phrases in modern English

• surface area — surface .area n the area of the outside of an object that can be measured …   Dictionary of contemporary English

• surface area — noun the extent of a 2 dimensional surface enclosed within a boundary (Freq. 2) the area of a rectangle it was about 500 square feet in area • Syn: ↑area, ↑expanse • Derivationally related forms: ↑areal ( …   Useful english dictionary

• surface area — UK / US noun [countable] Word forms surface area : singular surface area plural surface areas maths the total area of a surface or surfaces, especially the outside surfaces of an object enough paint to cover a surface area of 900 square feet …   English dictionary

• surface area — paviršiaus plotas statusas T sritis Standartizacija ir metrologija apibrėžtis Nagrinėjamojo paviršiaus plotas. atitikmenys: angl. surface area vok. Oberflächeninhalt, m rus. площадь поверхности, f pranc. aire de surface, f …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• surface area — paviršiaus plotas statusas T sritis fizika atitikmenys: angl. surface area vok. Oberflächeninhalt, m rus. площадь поверхности, f pranc. aire de surface, f …   Fizikos terminų žodynas

• surface area — noun The total area on the surface of a three dimensional figure …   Wiktionary

• surface area — Measure of the amount of surface presented by a catalyst particle to a reaction system. Many catalysts have surface areas in excess of 100 m2/gm …   Petroleum refining glossary

• surface area — noun (C) the area of the outside of an object that can be measured …   Longman dictionary of contemporary English

• surface pressure-surface area isotherm —  Surface Pressure Surface Area Isotherm  Изотерма поверхностное давление площадь   Зависимость поверхностного давления монослоя Ленгмюра на поверхности водной субфазы от площади, занимаемой молекулами поверхностно активного вещества в этом… …   Толковый англо-русский словарь по нанотехнологии. - М.