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The '''surface area''' of a solid object is the total area of the object's faces and curved surfaces. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of [[arc length]] of one-dimensional curves, or of the surface area for [[polyhedra]] (i.e., objects with flat polygonal [[Face (geometry)|faces]]), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a [[sphere]], are assigned surface area using their representation as [[parametric surface]]s. This definition of surface area is based on methods of [[infinitesimal]] [[calculus]] and involves [[partial derivative]]s and [[double integration]]. | |||
A general definition of surface area was sought by [[Henri Lebesgue]] and [[Hermann Minkowski]] at the turn of the twentieth century. Their work led to the development of [[geometric measure theory]], which studies various notions of surface area for irregular objects of any dimension. An important example is the [[Minkowski content]] of a surface. | |||
== | |||
While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical ''definition'' of area requires a great deal of care. | |||
This should provide a function | |||
: <math> S \mapsto A(S) </math> | |||
which assigns a positive [[real number]] to a certain class of [[surface]]s that satisfies several natural requirements. The most fundamental property of the surface area is its '''additivity''': ''the area of the whole is the sum of the areas of the parts''. More rigorously, if a surface ''S'' is a union of finitely many pieces ''S''<sub>1</sub>, …, ''S''<sub>''r''</sub> which do not overlap except at their boundaries, then | |||
: <math> A(S) = A(S_1) + \cdots + A(S_r). </math> | |||
Surface areas of flat polygonal shapes must agree with their geometrically defined [[area]]. Since surface area is a geometric notion, areas of [[congruence (geometry)|congruent]] surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the [[Euclidean group|group of Euclidean motions]]. These properties uniquely characterize surface area for a wide class of geometric surfaces called ''piecewise smooth''. Such surfaces consist of finitely many pieces that can be represented in the [[parametric surface|parametric form]] | |||
: <math> S_D: \vec{r}=\vec{r}(u,v), \quad (u,v)\in D </math> | |||
with a [[continuously differentiable]] function <math>\vec{r}.</math> The area of an individual piece is defined by the formula | |||
: <math> A(S_D) = \iint_D\left |\vec{r}_u\times\vec{r}_v\right | \, du \, dv. </math> | |||
Thus the area of ''S''<sub>''D''</sub> is obtained by integrating the length of the normal vector <math>\vec{r}_u\times\vec{r}_v</math> to the surface over the appropriate region ''D'' in the parametric ''uv'' plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs ''z'' = ''f''(''x'',''y'') and [[surface of revolution|surfaces of revolution]]. | |||
One of the subtleties of surface area, as compared to [[arc length]] of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by [[Hermann Schwarz]] that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area (Known as [[Schwarz's paradox]].) | |||
<ref name=sch1>http://www.math.usma.edu/people/Rickey/hm/CalcNotes/schwarz-paradox.pdf</ref> | |||
.<ref name=sch2>http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf</ref> | |||
Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by [[Henri Lebesgue]] and [[Hermann Minkowski]]. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of [[fractal]]s. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in [[geometric measure theory]]. A specific example of such an extension is the [[Minkowski content]] of the surface. | |||
== Common formulas == | |||
{| class="wikitable" | |||
|+ Surface areas of common solids | |||
|- | |||
!Shape | |||
!Equation | |||
!Variables | |||
|- | |||
|[[Cube]] | |||
|<math> 6s^2 \, </math> | |||
|''s'' = side length | |||
|- | |||
|[[Rectangular prism]] | |||
|<math> 2(\ell w + \ell h + wh) \, </math> | |||
|''ℓ'' = length, ''w'' = width, ''h'' = height | |||
|- | |||
|[[Triangular prism]] | |||
|<math> bh + l(a + b + c) </math> | |||
|''b'' = base length of triangle, ''h'' = height of triangle, ''l'' = distance between triangles, ''a'', ''b'', ''c'' = sides of triangle | |||
|- | |||
|All [[Prism (geometry)|Prisms]] | |||
|<math> 2B + Ph \, </math> | |||
|''B'' = the area of one base, ''P'' = the perimeter of one base, ''h'' = height | |||
|- | |||
|[[Sphere]] | |||
|<math> 4\pi r^2 = \pi d^2\, </math> | |||
|''r'' = radius of sphere, ''d'' = diameter | |||
|- | |||
|[[Spherical lune]] | |||
|<math> 2r^2\theta \, </math> | |||
|''r'' = radius of sphere, ''θ'' = [[dihedral angle]] | |||
|- | |||
|[[Torus]] | |||
|<math> (2\pi r)(2\pi R) = 4\pi^2 Rr</math> | |||
|''r'' = minor radius, ''R'' = major radius | |||
|- | |||
|Closed [[Cylinder (geometry)|cylinder]] | |||
|<math> 2\pi r^2 + 2\pi rh = 2\pi r(r+h) \, </math> | |||
|''r'' = radius of the circular base, ''h'' = height of the cylinder | |||
|- | |||
|Lateral surface area of a [[cone (geometry)|cone]] | |||
|<math> \pi r \left(\sqrt{r^2+h^2}\right) = \pi rs \, </math> | |||
|<math> s = \sqrt{r^2+h^2} </math><br> | |||
''s'' = slant height of the cone,<br> | |||
''r'' = radius of the circular base,<br> | |||
''h'' = height of the cone | |||
|- | |||
|Full surface area of a cone | |||
|<math> \pi r \left(r + \sqrt{r^2+h^2}\right) = \pi r(r + s) \, </math> | |||
| ''s'' = slant height of the cone,<br> | |||
''r'' = radius of the circular base,<br> | |||
''h'' = height of the cone | |||
|- | |||
|[[Pyramid (geometry)|Pyramid]] | |||
|<math>B + \frac{PL}{2}</math> | |||
|''B'' = area of base, ''P'' = perimeter of base, ''L'' = slant height | |||
|- | |||
|[[Square pyramid]] | |||
|<math> b^2 + 2bs </math> | |||
|''b'' = base length, ''s'' = slant height | |||
|} | |||
===Ratio of surface areas of a sphere and cylinder of the same Radius and Height=== | |||
[[Image:Inscribed cone sphere cylinder.svg|thumb|300px|A cone, sphere and cylinder of radius ''r'' and height ''h''.]] | |||
The below given formulas can be used to show that the surface area of a [[sphere]] and [[cylinder (geometry)|cylinder]] of the same radius and height are in the ratio '''2 : 3''', as follows. | |||
Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere). | |||
<math>\begin{array}{rlll} | |||
\text{Sphere surface area} & = 4 \pi r^2 & & = (2 \pi r^2) \times 2 \\ | |||
\text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3 | |||
\end{array}</math> | |||
The discovery of this ratio is credited to [[Archimedes]].<ref>{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title = Tomb of Archimedes: Sources|publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-01-02}}</ref> | |||
== In chemistry == | |||
{{see also|Accessible surface area}} | |||
Surface area is important in [[chemical kinetics]]. Increasing the surface area of a substance generally increases the [[reaction rate|rate]] of a [[chemical reaction]]. For example, [[iron]] in a fine powder will [[combustion|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 == | |||
{{see also|Surface-area-to-volume ratio}} | |||
[[Image:Mitochondrion 186.jpg|right|thumb|The inner membrane of the [[mitochondrion]] has a large surface area due to infoldings, allowing higher rates of [[cellular respiration]] (electron [[micrograph]]).]] | |||
The surface area of an organism is important in several considerations, such as regulation of body temperature and [[digestion]]. Animals use their [[teeth]] to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains [[microvilli]], greatly increasing the area available for absorption. [[Elephant]]s have large [[ear]]s, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss. | |||
The [[surface area to volume ratio]] (SA:V) of a [[cell (biology)|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. Indeed, representing a cell as an idealized [[sphere]] of radius ''r'', the volume and surface area are, respectively, ''V'' = 4/3 π ''r''<sup>3</sup>; ''SA'' = 4 π ''r''<sup>2</sup>. The resulting surface area to volume ratio is therefore 3/''r''. Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume. | |||
== References == | |||
<references /> | |||
* {{eom|title=Area|id=A/a013180|author=Yu.D. Burago, V.A. Zalgaller, L.D. Kudryavtsev}} | |||
==External links== | |||
*[http://blog.thinkwell.com/2010/07/6th-grade-math-surface-area.html Surface Area Video] at Thinkwell | |||
[[Category:Area]] | |||
[[es:Área de superficies]] | |||
[[pl:Pole powierzchni]] | |||
[[sv:Area]] | |||
Revision as of 00:20, 24 November 2012
The surface area of a solid object is the total area of the object's faces and curved surfaces. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.
==
While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function
which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S1, …, Sr which do not overlap except at their boundaries, then
Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of finitely many pieces that can be represented in the parametric form
with a continuously differentiable function The area of an individual piece is defined by the formula
Thus the area of SD is obtained by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f(x,y) and surfaces of revolution.
One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area (Known as Schwarz's paradox.) [1] .[2]
Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study 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 geometric measure theory. A specific example of such an extension is the Minkowski content of the surface.
Common formulas
| Shape | Equation | Variables |
|---|---|---|
| Cube | s = side length | |
| Rectangular prism | ℓ = length, w = width, h = height | |
| Triangular prism | b = base length of triangle, h = height of triangle, l = distance between triangles, a, b, c = sides of triangle | |
| All Prisms | B = the area of one base, P = the perimeter of one base, h = height | |
| Sphere | r = radius of sphere, d = diameter | |
| Spherical lune | r = radius of sphere, θ = dihedral angle | |
| Torus | r = minor radius, R = major radius | |
| Closed cylinder | r = radius of the circular base, h = height of the cylinder | |
| Lateral surface area of a cone | s = slant height of the cone, | |
| Full surface area of a cone | s = slant height of the cone, r = radius of the circular base, | |
| Pyramid | B = area of base, P = perimeter of base, L = slant height | |
| Square pyramid | b = base length, s = slant height |
Ratio of surface areas of a sphere and cylinder of the same Radius and Height
The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows.
Let the radius be r and the height be h (which is 2r for the sphere).
The discovery of this ratio is credited to Archimedes.[3]
In chemistry
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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
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We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.
The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.
Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.
is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease
In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.
The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion. Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli, greatly increasing the area available for absorption. Elephants have large ears, allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss.
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. Indeed, representing a cell as an idealized sphere of radius r, the volume and surface area are, respectively, V = 4/3 π r3; SA = 4 π r2. The resulting surface area to volume ratio is therefore 3/r. Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume.
References
External links
- Surface Area Video at Thinkwell