# Surface-area-to-volume ratio

(Redirected from Surface area to volume ratio)

For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is the sphere, 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. If a cell is too large, not enough materials will be able to cross the membrane fast enough to accommodate the high cellular volume.

## 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. Examples include grain dust; while grain isn't typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt.

High surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize free energy.

## Biology

The ratio between the surface area and volume of cells and organisms has an enormous impact on their biology (the physiology, behavior, and other qualities of a particular organism or class of organisms). 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.

An increased surface area to volume ratio also means increased exposure to the environment. The many tentacles of jellyfish and anemones are the result of increased surface area for the acquisition of food. Greater surface area allows more of the surrounding water to be sifted for food.

Individual organs in animals are often based on the principle of greater surface area. The lung is an organ with numerous internal branchings that increase the surface area through which oxygen is passed into the blood and carbon dioxide is released from the blood. The intestine has a finely wrinkled internal surface, increasing the area through which nutrients are absorbed by the body. This is done to increase the surface area in which diffusion of oxygen and carbon dioxide in the lungs and diffusion of nutrients in villi of the small intestine can occur.

Cells can achieve a high surface area to volume ratio by being long and thin (nerve cells) or convoluted (microvilli)

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.

The surface to volume ratios of organisms of different sizes also leads to some observations in biogeography such as Bergmann's rule.

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.

## Mathematical examples

Shape Characteristic Length ${\displaystyle a}$ Surface Area Volume SA/V ratio SA/V ratio for unit volume
Tetrahedron side ${\displaystyle {\sqrt {3}}a^{2}}$ ${\displaystyle {\frac {{\sqrt {2}}a^{3}}{12}}}$ ${\displaystyle {\frac {6{\sqrt {6}}}{a}}\approx {\frac {14.697}{a}}}$ 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 ${\displaystyle 4\pi a^{2}}$ ${\displaystyle {\frac {4\pi a^{3}}{3}}}$ ${\displaystyle {\frac {3}{a}}}$ 4.836
Example Table
side of cube side2 Area of side 6*side2 Area of Cube's Surface 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