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Surface-area-to-volume ratio
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<h2 id="for-solid-spheres">For solid spheres</h2> <p>A <i><a href="/facts/Solid_sphere/f9vpMVMp">solid sphere</a></i> or <a href="/facts/Ball_(mathematics)/f9vpMVMp"><i>ball</i></a> is a <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional</a> object, being the <a href="/facts/Solid_figure/tHH3JFoq">solid figure</a> bounded by a <a href="/facts/Sphere/jywYLNM2">sphere</a>. (In geometry, the term <i>sphere</i> properly refers only to the surface, so a sphere thus lacks <a href="/facts/Volume/aeAXCBUd">volume</a> in this context.) </p><p>For an ordinary three-dimensional ball, the SA:V can be calculated using the standard equations for the surface and volume, which are, respectively, S A = 4 π r 2 {\displaystyle SA=4\pi {r^{2}}} and V = ( 4 / 3 ) π r 3 {\displaystyle V=(4/3)\pi {r^{3}}} . For the unit case in which <i>r</i> = 1 the SA:V is thus 3. For the general case, SA:V equals 3/<i>r</i>, in an inverse relationship with the radius - if the radius is doubled, the SA:V halves (see figure). </p> <h3>For <i>n</i>-dimensional balls</h3> <p>Balls exist in any dimension and are generically called <i><a href="/facts/N-ball/f9vpMVMp"><i>n</i>-balls</a></i> or <i>hyperballs</i>, where <i>n</i> 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: </p> V = r n π n / 2 Γ ( 1 + n / 2 ) {\displaystyle V={\frac {r^{n}\pi ^{n/2}}{\Gamma (1+n/2)}}} S A = n r n − 1 π n / 2 Γ ( 1 + n / 2 ) {\displaystyle SA={\frac {nr^{n-1}\pi ^{n/2}}{\Gamma (1+{n/2})}}} <p>So the ratio equals S A / V = n r − 1 {\displaystyle 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. </p> <h2 id="dimension-and-units">Dimension and units</h2> <p>The surface-area-to-volume ratio has <a href="/facts/Physical_dimension/brTWIYNF">physical dimension</a> <a href="/facts/Inverse_length/mlrDdsFD">inverse length</a> (L−1) and is therefore expressed in <a href="/facts/Units_of_measurement/pvTpWbmC">units</a> of inverse metre (m−1) or its <a href="/facts/Unit_prefix/CTotM82r">prefixed</a> <a href="/facts/Unit_multiple/7d7Uqd9N">unit multiples</a> and submultiples. As an example, a <a href="/facts/Cube/65lUaAqN">cube</a> with sides of length 1 <a href="/facts/Centimeter/urPkBdGR">cm</a> will have a surface area of 6 cm2 and a volume of 1 cm3. The surface to volume ratio for this cube is thus </p> SA:V = 6 cm 2 1 cm 3 = 6 cm − 1 {\displaystyle {\mbox{SA:V}}={\frac {6~{\mbox{cm}}^{2}}{1~{\mbox{cm}}^{3}}}=6~{\mbox{cm}}^{-1}} . <p>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 <a href="/facts/Compactness_measure_of_a_shape/mhaoMU0V">compact</a> shape. </p> <h2 id="applications">Applications</h2> <h3>Physical chemistry</h3> <p class="note">See also: <a href="/facts/Dust_explosion/XKVeURog">Dust explosion</a></p> <p>Materials with high <a href="/facts/Surface_area/WNowzFCw">surface area</a> to <a href="/facts/Volume/aeAXCBUd">volume</a> ratio (e.g. very small diameter, very <a href="/facts/Porosity/UoyAJU8X">porous</a>, or otherwise not <a href="/facts/Compactness_measure_of_a_shape/mhaoMU0V">compact</a>) 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, <a href="/facts/Great_Mill_Disaster/HlPHPW7v">grain dust is explosive</a>. Finely ground salt dissolves much more quickly than coarse salt. </p><p>A high surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize <a href="/facts/Thermodynamic_free_energy/GtjtQqnq">free energy</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h3>Biology</h3> <p>The ratio between the surface area and volume of <a href="/facts/Cell_(biology)/bLM9UQvy">cells</a> and <a href="/facts/Organism/j8JkL09u">organisms</a> has an enormous impact on their <a href="/facts/Biology/1M1ee3pB">biology</a>, including their <a href="/facts/Physiology/fh25y2pQ">physiology</a> and <a href="/facts/Behavior/0Y1VqmVX">behavior</a>. 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. </p><p>An increased surface area to volume ratio also means increased exposure to the environment. The finely-branched appendages of <a href="/facts/Filter_feeder/hmVLwWB5">filter feeders</a> such as <a href="/facts/Krill/q11notjt">krill</a> provide a large surface area to sift the water for food.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>Individual organs like the <a href="/facts/Lung/gTWXI6Fj">lung</a> have numerous internal branchings that increase the surface area; in the case of the lung, the large surface supports gas exchange, bringing <a href="/facts/Oxygen/acyn6o8r">oxygen</a> into the blood and releasing <a href="/facts/Carbon_dioxide/xGzpppEp">carbon dioxide</a> from the blood.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Similarly, the <a href="/facts/Small_intestine/1AQ8RJ4m">small intestine</a> has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p><p>Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the <a href="/facts/Microvilli/tlI7DwCh">microvilli</a> lining the <a href="/facts/Small_intestine/1AQ8RJ4m">small intestine</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p><p>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. </p><p>The surface to volume ratios of organisms of different sizes also leads to some <a href="/facts/Biological_rule/6GohqdHP">biological rules</a> such as <a href="/facts/Allen%2527s_rule/vFVUN3Sa">Allen's rule</a>, <a href="/facts/Bergmann%2527s_rule/3x5kMnFR">Bergmann's rule</a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> and <a href="/facts/Gigantothermy/xIVetrMI">gigantothermy</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> <h3>Fire spread</h3> <p>In the context of <a href="/facts/Wildfire/LU3797A0">wildfires</a>, 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. </p> <h3>Planetary cooling</h3> <p>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 <a href="/facts/4_Vesta/xnQ8acw1">Vesta</a> (r=263 km), the ratio is so high that astronomers were surprised to find that it <i>did</i> differentiate and have brief volcanic activity. The <a href="/facts/Moon/zKHqEMPb">moon</a>, <a href="/facts/Mercury_(planet)/lVa1GIyX">Mercury</a> and <a href="/facts/Mars/AQGtVvz0">Mars</a> 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.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> <a href="/facts/Venus_(planet)/pADnueP6">Venus</a> and <a href="/facts/Earth/HLLmDfj0">Earth</a> (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.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> <h2 id="mathematical-examples">Mathematical examples</h2> <table><tbody><tr><th>Shape</th><th>Image</th><th>Characteristiclength a {\displaystyle a} </th><th>SA/V ratio</th><th>SA/V ratio forunit volume</th></tr><tr><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a></td><td></td><td>edge</td><td> 6 6 a ≈ 14.697 a {\displaystyle {\frac {6{\sqrt {6}}}{a}}\approx {\frac {14.697}{a}}} </td><td>7.21</td></tr><tr><td><a href="/facts/Cube/65lUaAqN">Cube</a></td><td></td><td>edge</td><td> 6 a {\displaystyle {\frac {6}{a}}} </td><td>6</td></tr><tr><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a></td><td></td><td>edge</td><td> 3 6 a ≈ 7.348 a {\displaystyle {\frac {3{\sqrt {6}}}{a}}\approx {\frac {7.348}{a}}} </td><td>5.72</td></tr><tr><td><a href="/facts/Dodecahedron/7r519eCl">Dodecahedron</a></td><td></td><td>edge</td><td> 12 25 + 10 5 ( 15 + 7 5 ) a ≈ 2.694 a {\displaystyle {\frac {12{\sqrt {25+10{\sqrt {5}}}}}{(15+7{\sqrt {5}})a}}\approx {\frac {2.694}{a}}} </td><td>5.31</td></tr><tr><td><a href="/facts/Capsule_(geometry)/j1BDCfbH">Capsule</a></td><td></td><td>radius (R)</td><td> 12 5 a {\displaystyle {\frac {12}{5a}}} </td><td>5.251</td></tr><tr><td><a href="/facts/Icosahedron/wWbpnDYp">Icosahedron</a></td><td></td><td>edge</td><td> 12 3 ( 3 + 5 ) a ≈ 3.970 a {\displaystyle {\frac {12{\sqrt {3}}}{(3+{\sqrt {5}})a}}\approx {\frac {3.970}{a}}} </td><td>5.148</td></tr><tr><td><a href="/facts/Sphere/jywYLNM2">Sphere</a></td><td></td><td>radius</td><td> 3 a {\displaystyle {\frac {3}{a}}} </td><td>4.83598</td></tr></tbody></table> Examples of cubes of different sizes<table><tbody><tr><th>Side ofcube</th><th>Side2</th><th>Area of asingle face</th><th>6 × side2</th><th>Area ofentire cube(6 faces)</th><th>Side3</th><th>Volume</th><th>Ratio ofsurface areato volume</th></tr><tr><td>2</td><td>2×2</td><td>4</td><td>6×2×2</td><td>24</td><td>2×2×2</td><td>8</td><td>3:1</td></tr><tr><td>4</td><td>4×4</td><td>16</td><td>6×4×4</td><td>96</td><td>4×4×4</td><td>64</td><td>3:2</td></tr><tr><td>6</td><td>6×6</td><td>36</td><td>6×6×6</td><td>216</td><td>6×6×6</td><td>216</td><td>3:3</td></tr><tr><td>8</td><td>8×8</td><td>64</td><td>6×8×8</td><td>384</td><td>8×8×8</td><td>512</td><td>3:4</td></tr><tr><td>12</td><td>12×12</td><td>144</td><td>6×12×12</td><td>864</td><td>12×12×12</td><td>1,728</td><td>3:6</td></tr><tr><td>20</td><td>20×20</td><td>400</td><td>6×20×20</td><td>2,400</td><td>20×20×20</td><td>8,000</td><td>3:10</td></tr><tr><td>50</td><td>50×50</td><td>2,500</td><td>6×50×50</td><td>15,000</td><td>50×50×50</td><td>125,000</td><td>3:25</td></tr><tr><td>1,000</td><td>1,000×1,000</td><td>1,000,000</td><td>6×1,000×1,000</td><td>6,000,000</td><td>1,000×1,000×1,000</td><td>1,000,000,000</td><td>3:500</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Compactness_measure_of_a_shape/mhaoMU0V">Compactness measure of a shape</a></li> <li><a href="/facts/Dust_explosion/XKVeURog">Dust explosion</a></li> <li><a href="/facts/On_Being_the_Right_Size/xtthLCcE">On Being the Right Size</a></li> <li><a href="/facts/Square%25E2%2580%2593cube_law/nq45ymy6">Square–cube law</a></li> <li><a href="/facts/Specific_surface_area/l8Dhv2S5">Specific surface area</a></li></ul> <ul><li>Schmidt-Nielsen, Knut (1984). <a href="https://archive.org/details/scalingwhyisanim0000schm"><i>Scaling: Why is Animal Size so Important?</i></a>. New York, NY: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-26657-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/10697247">10697247</a>.</li> <li>Vogel, Steven (1988). <i>Life's Devices: The Physical World of Animals and Plants</i>. Princeton, NJ: Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-08504-3. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/18070616">18070616</a>.</li></ul> Specific <h2 id="external-links">External links</h2> <ul><li><a href="http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html">Sizes of Organisms: The Surface Area:Volume Ratio</a> <a href="https://web.archive.org/web/20170814195742/http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html">Archived</a> 2017-08-14 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li> <li><a href="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</a></li> <li><a href="https://web.archive.org/web/20140521215017/http://www.nwcg.gov/pms/RxFire/FEG.pdf">Previous link not working, references are in this document, PDF</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Planinšič, Gorazd; Vollmer, Michael (February 20, 2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics. 29 (2): 369–384. Bibcode:2008EJPh...29..369P. doi:10.1088/0143-0807/29/2/017. S2CID 55488270. Retrieved 9 July 2021. <a href="https://iopscience.iop.org/article/10.1088/0143-0807/29/2/017/meta" target="_blank">https://iopscience.iop.org/article/10.1088/0143-0807/29/2/017/meta</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Planinšič, Gorazd (2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics European Physical Society, Find Out More. 29 (2): 369–384. Bibcode:2008EJPh...29..369P. doi:10.1088/0143-0807/29/2/017. 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"Surface Area to Volume Ratio: A Natural Variable for Bacterial Morphogenesis". Trends in Microbiology. 26 (10): 815–832. doi:10.1016/j.tim.2018.04.008. PMC 6150810. PMID 29843923. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6150810" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6150810</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Louw, Gideon N. (1993). Physiological Animal Ecology. Longman Pub Group. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Nguyen, Thanh Danh; Olufemi E., Kadri; Vassilios I., Sikavitsas; Voronov, Roman S. (2019). "Scaffolds with a High Surface Area-to-Volume Ratio and Cultured Under Fast Flow Perfusion Result in Optimal O2 Delivery to the Cells in Artificial Bone Tissues". 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