View source for Surface area

{{Short description|Measure of a two-dimensional surface}}
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{{EngvarB|date=June 2022}}
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[[Image:Sphere wireframe 10deg 6r.svg|right|thumb|A [[sphere]] of radius {{mvar|r}} has surface area {{math|4''&pi;r''<sup>2</sup>}}.]]

The '''surface area''' (symbol '''''A''''') of a [[Solid geometry|solid]] object is a measure of the total [[area]] that the [[Surface (mathematics)|surface]] of the object occupies.<ref>{{MathWorld|title=Surface Area|urlname=SurfaceArea}}</ref> 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.

==Definition==
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 (topology)|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]].

[[File:Schwarz-lantern.gif|thumb|[[Schwarz lantern]] with <math>M</math> axial slices and <math>N</math> radial vertices. The limit of the area as <math>M</math> and <math>N</math> tend to infinity doesn't converge. In particular it doesn't converge to the area of the cylinder.]]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; this example is known as the [[Schwarz lantern]].<ref name=sch1>{{cite web|url=http://fredrickey.info/hm/CalcNotes/schwarz-paradox.pdf|title=Schwarz's Paradox|access-date=2017-03-21|url-status=live|archive-url=https://web.archive.org/web/20160304073957/http://fredrickey.info/hm/CalcNotes/schwarz-paradox.pdf|archive-date=2016-03-04}}</ref><ref name=sch2>{{cite web |url=http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf |title=Archived copy |access-date=2012-07-24 |url-status=dead |archive-url=https://web.archive.org/web/20111215152255/http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf |archive-date=2011-12-15 }}</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 ==
{{anchor|List of surface area formulas}}
{{See also|List of formulas in elementary geometry}}

{| class="wikitable"
|+ Surface areas of common solids
|-
!Shape
!Formula/Equation
!Variables
|-
|[[Cube]]
|<math> 6a^2 </math>
|''a'' = side length
|-
|[[Cuboid]]
|<math> 2\left(lb+lh+bh\right) </math>
|''l'' = length, ''b'' = breadth, ''h'' = height
|-
|[[Triangular prism]]
|<math> bh+l\left(p+q+r\right) </math>
|''b'' = base length of triangle, ''h'' = height of triangle, ''l'' = distance between triangular bases, ''p'', ''q'', ''r'' = 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
|-
|Hemisphere
|<math> 2\pi r^2 </math> 
|''r'' = radius of the hemisphere
|-
|Hemispherical shell
|<math> \pi \left(3R^2+r^2\right) </math>
|''R'' = external radius of hemisphere, ''r'' = internal radius of hemisphere
|-
|[[Spherical lune]]
|<math> 2r^2\theta </math>
|''r'' = radius of sphere, ''θ'' = [[dihedral angle]]
|-
|[[Torus]]
|<math> \left(2\pi r\right)\left(2\pi R\right)=4\pi^2Rr</math>
|''r'' = minor radius (radius of the tube), ''R'' = major radius (distance from center of tube to center of torus)
|-
|Closed [[Cylinder (geometry)|cylinder]]
|<math> 2\pi r^2+2\pi rh=2\pi r\left(r+h\right) </math>
|''r'' = radius of the circular base, ''h'' = height of the cylinder
|-
|Cylindrical [[Annulus (mathematics)|annulus]]
|<math> 2\pi Rh+2\pi rh+2(\pi R^2-\pi r^2) =2\pi (R+r)(R-r+h) </math>
|''R'' = External radius
''r'' = Internal radius, ''h'' = height
|-
|[[Capsule (geometry)|Capsule]]
|<math> 2\pi r(2r+h)  </math>
|''r'' = radius of the hemispheres and cylinder, ''h'' = height of the cylinder
|-
|Curved surface area of a [[cone (geometry)|cone]]
|<math> \pi r\sqrt{r^2+h^2}=\pi rs </math>
|<math> s=\sqrt{r^2+h^2} </math><br/>
''s'' = slant height of the cone, ''r'' = radius of the circular base, ''h'' = height of the cone
|-
|Full surface area of a cone
|<math> \pi r\left(r+\sqrt{r^2+h^2}\right)=\pi r\left(r +s\right) </math>
| ''s'' = slant height of the cone, ''r'' = radius of the circular base, ''h'' = height of the cone
|-
|Regular [[Pyramid (geometry)|Pyramid]]
|<math>B+\frac{Ps}{2}</math>
|''B'' = area of base, ''P'' = perimeter of base, ''s'' = slant height
|-
|[[Square pyramid]]
|<math> b^2 + 2bs = b^2+ 2b\sqrt{\left(\frac{b}{2}\right)^2+h^2} </math>
|''b'' = base length, ''s'' = slant height, ''h'' = vertical height
|-
|Rectangular pyramid
|<math> lb+l\sqrt{\left(\frac{b}{2}\right)^2+h^2}+ b\sqrt{\left(\frac{l}{2}\right)^2+h^2} </math>
|''l'' = length, ''b'' = breadth, ''h'' = height
|-
|[[Tetrahedron]]
|<math> \sqrt{3}a^2 </math>
|''a'' = side length
|-
|[[Surface of revolution]]
|<math>2\pi \int_a^b {f(x) \sqrt{1+(f'(x))^2} dx}</math>
|
|-
|[[Parametric surface]]
|<math>\iint_D \left \vert \vec{r}_u \times \vec{r}_v \right \vert dA</math>
|<math>\vec{r}</math> = parametric vector equation of surface,

<math>\vec{r}_u</math> = partial derivative of <math>\vec{r}</math> with respect to <math>u</math>,<br/>
<math>\vec{r}_v</math> = partial derivative of <math>\vec{r}</math> with respect to <math>v</math>,<br/>
<math>D</math> = shadow region
|}

===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&nbsp;:&nbsp;3''', as follows.

Let the radius be ''r'' and the height be ''h'' (which is 2''r'' for the sphere).

<math display="block">\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|access-date = 2007-01-02|url-status = live|archive-url = https://web.archive.org/web/20061209201723/http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|archive-date = 2006-12-09}}</ref>
{{Clear}}

== In chemistry ==
[[File:Surface area.svg|thumb|Surface area of particles of different sizes.]]
{{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]],<ref>{{Cite journal |last1=Nasr |first1=Somaye |last2=Plucknett |first2=Kevin P. |date=2014-02-20 |title=Kinetics of Iron Ore Reduction by Methane for Chemical Looping Combustion |url=https://pubs.acs.org/doi/10.1021/ef402142q |journal=Energy & Fuels |language=en |volume=28 |issue=2 |pages=1387–1395 |doi=10.1021/ef402142q |issn=0887-0624}}</ref> while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.
{{Clear}}

== In biology ==
[[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]]).<ref>{{Cite journal |last1=Paumard |first1=Patrick |last2=Vaillier |first2=Jacques |last3=Coulary |first3=Bénédicte |last4=Schaeffer |first4=Jacques |last5=Soubannier |first5=Vincent |last6=Mueller |first6=David M. |last7=Brèthes |first7=Daniel |last8=di Rago |first8=Jean-Paul |last9=Velours |first9=Jean |date=2002-02-01 |title=The ATP synthase is involved in generating mitochondrial cristae morphology |url=http://emboj.embopress.org/cgi/doi/10.1093/emboj/21.3.221 |journal=The EMBO Journal |language=en |volume=21 |issue=3 |pages=221–230 |doi=10.1093/emboj/21.3.221 |pmc=125827 |pmid=11823415}}</ref>]]
The surface area of an organism is important in several considerations, such as regulation of body temperature and [[digestion]].<ref>{{Cite journal |last=Narasimhan |first=Arunn |date=2008-07-01 |title=Why do elephants have big ear flaps? |url=https://doi.org/10.1007/s12045-008-0070-5 |journal=Resonance |language=en |volume=13 |issue=7 |pages=638–647 |doi=10.1007/s12045-008-0070-5 |issn=0973-712X}}</ref> Animals use their [[teeth]] to grind food down into smaller particles, increasing the surface area available for digestion.<ref>{{Citation |last=Feher |first=Joseph |title=Mouth and Esophagus |date=2012 |work=Quantitative Human Physiology |pages=689–700 |url=https://doi.org/10.1016/B978-0-12-382163-8.00077-3 |access-date=2024-03-30 |publisher=Elsevier |doi=10.1016/b978-0-12-382163-8.00077-3|isbn=978-0-12-382163-8 }}</ref> The epithelial tissue lining the digestive tract contains [[microvilli]], greatly increasing the area available for absorption.<ref>{{Cite web |title=Microvillus {{!}} Description, Anatomy, & Function {{!}} Britannica |url=https://www.britannica.com/science/microvillus |access-date=2024-03-30 |website=www.britannica.com |language=en}}</ref> [[Elephant]]s have large [[ear]]s, allowing them to regulate their own body temperature.<ref>{{Cite journal |last=Wright |first=P. G. |date=1984 |title=Why do elephants flap their ears? |url=https://www.ajol.info/index.php/az/article/view/152820 |journal=African Zoology |language=en |volume=19 |issue=4 |pages=266–269 |issn=2224-073X}}</ref> In other instances, animals will need to minimize surface area;<ref>{{Cite journal |last1=Stocks |first1=Jodie M. |last2=Taylor |first2=Nigel A.S. |last3=Tipton |first3=Michael J. |last4=Greenleaf |first4=John E. |date=2004-05-01 |title=Human Physiological Responses to Cold Exposure |url=https://www.ingentaconnect.com/content/asma/asem/2004/00000075/00000005/art00011 |journal=Aviation, Space, and Environmental Medicine |volume=75 |issue=5 |pages=444–457|pmid=15152898 }}</ref> 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.<ref>{{Cite journal |last=Deaver |first=James R. |date=1978-11-01 |title=Modeling Limits to Cell Size |url=https://online.ucpress.edu/abt/article/40/8/502/11842/Modeling-Limits-to-Cell-Size |journal=The American Biology Teacher |language=en |volume=40 |issue=8 |pages=502–504 |doi=10.2307/4446369 |jstor=4446369 |issn=0002-7685}}</ref> Indeed, representing a cell as an idealized [[sphere]] of radius {{mvar|r}}, the volume and surface area are, respectively, {{math|1=''V'' = (4/3)''πr''<sup>3</sup>}} and {{math|1=''SA'' = 4''πr''<sup>2</sup>}}. The resulting surface area to volume ratio is therefore {{math|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.

== See also ==
* [[Perimeter length]]
* [[Projected area]]
* [[BET theory]], technique for the measurement of the specific surface area of materials
* [[Spherical area]]
* [[Surface integral]]

== References ==

{{Reflist}}

* {{eom|title=Area|id=A/a013180|author=Yu.D. Burago|author2=V.A. Zalgaller|author3=L.D. Kudryavtsev}}

==External links==
*[http://blog.thinkwell.com/2010/07/6th-grade-math-surface-area.html Surface Area Video] at Thinkwell

[[Category:Area]]