surface-area-to-volume ratio

File:SAV n3.png

A solid sphere or ball is a three-dimensional object, being the solid figure bounded by a sphere. (In geometry, the term sphere properly refers only to the surface, so a sphere thus lacks volume in this context.)

For an ordinary three-dimensional ball, the SA:V can be calculated using the standard equations for the surface and volume, which are, respectively, $SA=4\backslash pi\{r^2\}$ and $V=(4/3)\backslash pi\{r^3\}$. For the unit case in which r = 1 the SA:V is thus 3. For the general case, SA:V equals 3/r, in an inverse relationship with the radius - if the radius is doubled, the SA:V halves (see figure).

Balls exist in any dimension and are generically called n-balls or hyperballs, where n is the number of dimensions.

The same reasoning can be generalized to n-balls using the general equations for volume and surface area, which are:

:$V=\backslash frac\{r^n\backslash pi^\{n/2\}\}\{\backslash Gamma(1+n/2)\}$

:$SA=\backslash frac\{nr^\{n-1\}\backslash pi^\{n/2\}\}\{\backslash Gamma(1+\{n/2\})\}$

So the ratio equals $SA/V=nr^\{-1\}$. Thus, the same linear relationship between area and volume holds for any number of dimensions (see figure): doubling the radius always halves the ratio.

The surface-area-to-volume ratio has physical dimension inverse length (L⁻¹) and is therefore expressed in units of inverse metre (m⁻¹) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm² and a volume of 1 cm³. The surface to volume ratio for this cube is thus

:$\backslash mbox\{SA:V\}\; =\; \backslash frac\{6~\backslash mbox\{cm\}^2\}\{1~\backslash mbox\{cm\}^3\}\; =\; 6~\backslash mbox\{cm\}^\{-1\}$.

For a given shape, SA:V is inversely proportional to size. A cube 2 cm on a side has a ratio of 3 cm⁻¹, half that of a cube 1 cm on a side. Conversely, preserving SA:V as size increases requires changing to a less compact shape.

{{unreferenced section|date=February 2014}}

{{see also|Dust explosion}}

Materials with high surface area to volume ratio (e.g. very small diameter, very porous, or otherwise not compact) react at much faster rates than monolithic materials, because more surface is available to react. An example is grain dust: while grain is not typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt.

A high surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize free energy.{{Cite journal |last=Whittingham |first=M |date=May 1989 |title=Basic solid state chemistry Anthony R. West, John Wiley & Sons, New York (1988), 415 pages £13.95 ($32.95) |url=https://doi.org/10.1016/0167-2738(89)90043-x |journal=Solid State Ionics |volume=34 |issue=3 |pages=213 |doi=10.1016/0167-2738(89)90043-x |issn=0167-2738|url-access=subscription }}

File:Human jejunum microvilli 2 - TEM.jpg lining the small intestine increase the surface area over which they can absorb nutrients with a carpet of tuftlike microvilli.]]

The ratio between the surface area and volume of cells and organisms has an enormous impact on their biology, including their physiology and behavior. For example, many aquatic microorganisms have increased surface area to increase their drag in the water. This reduces their rate of sink and allows them to remain near the surface with less energy expenditure.{{citation needed|date=January 2018}}

An increased surface area to volume ratio also means increased exposure to the environment. The finely-branched appendages of filter feeders such as krill provide a large surface area to sift the water for food.Kils, U.: Swimming and feeding of Antarctic Krill, Euphausia superba - some outstanding energetics and dynamics - some unique morphological details. In Berichte zur Polarforschung, Alfred Wegener Institute for Polar and Marine Research, Special Issue 4 (1983): "On the biology of Krill Euphausia superba", Proceedings of the Seminar and Report of Krill Ecology Group, Editor S. B. Schnack, 130-155 and title page image.

Individual organs like the lung have numerous internal branchings that increase the surface area; in the case of the lung, the large surface supports gas exchange, bringing oxygen into the blood and releasing carbon dioxide from the blood.{{cite book |last1= Tortora |first1= Gerard J. |last2=Anagnostakos|first2=Nicholas P.| title=Principles of anatomy and physiology |url= https://archive.org/details/principlesofan1987tort |url-access= registration |pages=[https://archive.org/details/principlesofan1987tort/page/556 556–582]|edition= Fifth |location= New York |publisher= Harper & Row, Publishers|date= 1987 |isbn= 978-0-06-350729-6 }}{{cite book |last1=Williams |first1=Peter L |last2=Warwick |first2=Roger |last3=Dyson|first3=Mary |last4=Bannister |first4=Lawrence H. |title=Gray's Anatomy| pages=1278–1282 |location=Edinburgh|publisher=Churchill Livingstone | edition=Thirty-seventh |date=1989|isbn= 0443-041776 }} Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.{{cite book |author=Romer, Alfred Sherwood|author2=Parsons, Thomas S.|year=1977 |title=The Vertebrate Body |publisher=Holt-Saunders International |location= Philadelphia, PA|pages= 349–353|isbn= 978-0-03-910284-5}}

Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine.{{cite book|author=Krause J. William|title=Krause's Essential Human Histology for Medical Students|url=https://books.google.com/books?id=cRayoldYrcUC&pg=PA37|access-date=25 November 2010|date=July 2005|publisher=Universal-Publishers|isbn=978-1-58112-468-2|pages=37–}}

Increased surface area can also lead to biological problems. More contact with the environment through the surface of a cell or an organ (relative to its volume) increases loss of water and dissolved substances. High surface area to volume ratios also present problems of temperature control in unfavorable environments.{{citation needed|date=January 2018}}

The surface to volume ratios of organisms of different sizes also leads to some biological rules such as Allen's rule, Bergmann's rule{{Cite journal | doi=10.1046/j.1365-2699.2003.00837.x | title=On the validity of Bergmann's rule | journal=Journal of Biogeography | volume=30 | issue=3 | pages=331–351 | date=2003-03-20 | last1=Meiri | first1=S. | last2=Dayan | first2=T. | bibcode=2003JBiog..30..331M | s2cid=11954818 }}{{cite journal | jstor=10.1086/303400 | title=Is Bergmann's Rule Valid for Mammals? | date=October 2000 | first1=Kyle G. | last1=Ashton | last2=Tracy | first2=Mark C. | last3=Queiroz | first3=Alan de | journal=The American Naturalist | volume=156 | issue=4 | pages=390–415 | doi=10.1086/303400| pmid=29592141 | bibcode=2000ANat..156..390A | s2cid=205983729 }}{{cite journal | title=Ecotypic variation in the context of global climate change: Revisiting the rules | first1=Virginie | last1=Millien | last2=Lyons | first2=S. Kathleen | last3=Olson | first3=Link |display-authors=etal | journal=Ecology Letters | date=May 23, 2006 | volume=9 | issue=7 | pages=853–869 | doi=10.1111/j.1461-0248.2006.00928.x | pmid=16796576 | doi-access= | bibcode=2006EcolL...9..853M }} and gigantothermy.{{cite web |url=http://www.bio.davidson.edu/people/midorcas/animalphysiology/websites/2005/Fitzpatrick/Gigantothermy.htm |archive-url=https://archive.today/20120630084621/http://www.bio.davidson.edu/people/midorcas/animalphysiology/websites/2005/Fitzpatrick/Gigantothermy.htm |url-status=dead |archive-date=2012-06-30 |title=Gigantothermy |year=2005 |first=Katie |last=Fitzpatrick |publisher=Davidson College |access-date=2011-12-21 }}

In the context of wildfires, the ratio of the surface area of a solid fuel to its volume is an important measurement. Fire spread behavior is frequently correlated to the surface-area-to-volume ratio of the fuel (e.g. leaves and branches). The higher its value, the faster a particle responds to changes in environmental conditions, such as temperature or moisture. Higher values are also correlated to shorter fuel ignition times, and hence faster fire spread rates.

A body of icy or rocky material in outer space may, if it can build and retain sufficient heat, develop a differentiated interior and alter its surface through volcanic or tectonic activity. The length of time through which a planetary body can maintain surface-altering activity depends on how well it retains heat, and this is governed by its surface area-to-volume ratio. For Vesta (r=263 km), the ratio is so high that astronomers were surprised to find that it did differentiate and have brief volcanic activity. The moon, Mercury and Mars have radii in the low thousands of kilometers; all three retained heat well enough to be thoroughly differentiated although after a billion years or so they became too cool to show anything more than very localized and infrequent volcanic activity. As of April 2019, however, NASA has announced the detection of a "marsquake" measured on April 6, 2019, by NASA's InSight lander.{{Cite web | url=https://www.space.com/insight-mars-lander-first-marsquake.html |title = Marsquake! NASA's InSight Lander Feels Its 1st Red Planet Tremor|website = Space.com|date = 23 April 2019}} Venus and Earth (r>6,000 km) have sufficiently low surface area-to-volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal.{{Cite web |url=http://www.astro.uvic.ca/~venn/A201/maths.6.planetary_cooling.pdf |title=Archived copy |access-date=2018-08-22 |archive-date=2018-06-13 |archive-url=https://web.archive.org/web/20180613195628/http://www.astro.uvic.ca/~venn/A201/maths.6.planetary_cooling.pdf |url-status=dead }}

{{anchor|List of surface-area-to-volume ratios}}

• Compactness measure of a shape
• Dust explosion
• On Being the Right Size
• Square–cube law
• Specific surface area

• {{cite book |last=Schmidt-Nielsen |first=Knut |year=1984 |title=Scaling: Why is Animal Size so Important? |publisher=Cambridge University Press |location=New York, NY |isbn=978-0-521-26657-4 |oclc=10697247 |url-access=registration |url=https://archive.org/details/scalingwhyisanim0000schm }}
• {{cite book |last=Vogel |first=Steven |year=1988 |title=Life's Devices: The Physical World of Animals and Plants |publisher=Princeton University Press |location=Princeton, NJ |isbn=978-0-691-08504-3 |oclc=18070616}}

;Specific

{{Reflist|colwidth=30em}}

• [http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html Sizes of Organisms: The Surface Area:Volume Ratio] {{Webarchive|url=https://web.archive.org/web/20170814195742/http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html |date=2017-08-14 }}
• [https://web.archive.org/web/20140310230709/http://www.nwcg.gov/var/glossary/surface-area-to-volume-ratio National Wildfire Coordinating Group: Surface Area to Volume Ratio]
• [https://web.archive.org/web/20140521215017/http://www.nwcg.gov/pms/RxFire/FEG.pdf Previous link not working, references are in this document, PDF]

Category:Chemical kinetics

Category:Cell biology

Category:Physiology

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