# Surface-area-to-volume ratio

The surface-area-to-volume ratio, also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. In chemical reactions involving a solid material, the surface area to volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed.

Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.

For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a ball, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with tiny spikes will have very large surface area for a given volume.

## SA:V for balls and N-balls

A ball is a three-dimensional object, being the filled-in version of a sphere ("sphere" properly refers only to the surface and a sphere thus has no volume). Balls exist in any dimension and are generically called n-balls, where n is the number of dimensions.

Plot of the surface-area:volume ratio (SA:V) for a 3-dimensional ball, showing the ratio decline inversely as the radius of the ball increases.

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

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

volume = ${\displaystyle r^{n}\pi ^{n/2} \over \Gamma (1+n/2)}$ ; surface area = ${\displaystyle nr^{n-1}\pi ^{n/2} \over \Gamma (1+{n/2})}$

Plot of surface-area:volume ratio (SA:V) for n-balls as a function of the number of dimensions and of radius size. Note the linear scaling as a function of dimensionality and the inverse scaling as a function of radius.

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

## Dimension

The surface-area-to-volume ratio has physical dimension L−1 (inverse length) and is therefore expressed in units of inverse distance. 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

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

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.

## Physical chemistry

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.

## Biology

Cells 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]

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.[1]

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.[2][3] Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.[4]

Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine.[5]

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]

The surface to volume ratios of organisms of different sizes also leads to some biological rules such as Allen's rule, Bergmann's rule[6][7][8] and gigantothermy.[9]

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.

## Planetary cooling

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.[10] 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.[11]

## Mathematical examples

Shape Characteristic
Length ${\displaystyle a}$
Surface Area Volume SA/V ratio SA/V ratio for
unit volume
Tetrahedron   edge ${\displaystyle {\sqrt {3}}a^{2}}$  ${\displaystyle {\frac {{\sqrt {2}}a^{3}}{12}}}$  ${\displaystyle {\frac {6{\sqrt {6}}}{a}}\approx {\frac {14.697}{a}}}$  7.21
Cube   side ${\displaystyle 6a^{2}}$  ${\displaystyle a^{3}}$  ${\displaystyle {\frac {6}{a}}}$  6
Octahedron   side ${\displaystyle 2{\sqrt {3}}a^{2}}$  ${\displaystyle {\frac {1}{3}}{\sqrt {2}}a^{3}}$  ${\displaystyle {\frac {3{\sqrt {6}}}{a}}\approx {\frac {7.348}{a}}}$  5.72
Dodecahedron   side ${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}a^{2}}$  ${\displaystyle {\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}$  ${\displaystyle {\frac {12{\sqrt {25+10{\sqrt {5}}}}}{(15+7{\sqrt {5}})a}}\approx {\frac {2.694}{a}}}$  5.31
Capsule   radius (R) ${\displaystyle 4\pi a^{2}+2\pi a*2a=8\pi a^{2}}$  ${\displaystyle {\frac {4\pi a^{3}}{3}}+\pi a^{2}*2a={\frac {10\pi a^{3}}{3}}}$  ${\displaystyle {\frac {12}{5a}}}$  5.251
Icosahedron   side ${\displaystyle 5{\sqrt {3}}a^{2}}$  ${\displaystyle {\frac {5}{12}}(3+{\sqrt {5}})a^{3}}$  ${\displaystyle {\frac {12{\sqrt {3}}}{(3+{\sqrt {5}})a}}\approx {\frac {3.970}{a}}}$  5.148
Sphere   radius ${\displaystyle 4\pi a^{2}}$  ${\displaystyle {\frac {4\pi a^{3}}{3}}}$  ${\displaystyle {\frac {3}{a}}}$  4.83598
Examples of cubes of different sizes
Side of
cube
Side2 Area of a
single face
6 × side2 Area of
entire cube
(6 faces)
Side3 Volume Ratio of
surface area
to volume
2 2x2 4 6x2x2 24 2x2x2 8 3:1
4 4x4 16 6x4x4 96 4x4x4 64 3:2
6 6x6 36 6x6x6 216 6x6x6 216 3:3
8 8x8 64 6x8x8 384 8x8x8 512 3:4
12 12x12 144 6x12x12 864 12x12x12 1728 3:6
20 20x20 400 6x20x20 2400 20x20x20 8000 3:10
50 50x50 2500 6x50x50 15000 50x50x50 125000 3:25
1000 1000x1000 1000000 6x1000x1000 6000000 1000x1000x1000 1000000000 3:500

## References

• Schmidt-Nielsen, Knut (1984). Scaling: Why is Animal Size so Important?. New York, NY: Cambridge University Press. ISBN 978-0-521-26657-4. OCLC 10697247.
• Vogel, Steven (1988). Life's Devices: The Physical World of Animals and Plants. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08504-3. OCLC 18070616.
Specific
1. ^ 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.
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6. ^ Meiri, S.; Dayan, T. (2003-03-20). "On the validity of Bergmann's rule". Journal of Biogeography. 30 (3): 331–351. doi:10.1046/j.1365-2699.2003.00837.x.
7. ^ Ashton, Kyle G.; Tracy, Mark C.; Queiroz, Alan de (October 2000). "Is Bergmann's Rule Valid for Mammals?". The American Naturalist. 156 (4): 390–415. doi:10.1086/303400. JSTOR 10.1086/303400. PMID 29592141. S2CID 205983729.
8. ^ Millien, Virginie; Lyons, S. Kathleen; Olson, Link; et al. (May 23, 2006). "Ecotypic variation in the context of global climate change: Revisiting the rules". Ecology Letters. 9 (7): 853–869. doi:10.1111/j.1461-0248.2006.00928.x. PMID 16796576.
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11. ^ http://www.astro.uvic.ca/~venn/A201/maths.6.planetary_cooling.pdf