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 surfaceareatovolume ratio decreases with increasing volume.
The surfaceareatovolume ratio, also called the surfacetovolume 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.
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.
Contents

Dimension 1

Physical chemistry 2

Biology 3

Fire spread 4

Mathematical examples 5

See also 6

References 7

External links 8

Further reading 9
Dimension
The surfaceareatovolume 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 cm^{2} and a volume of 1 cm^{3}. The surface to volume ratio for this cube is thus

\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 get around having 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.
Fire spread
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 surfaceareatovolume 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 a

Surface Area

Volume

SA/V ratio

SA/V ratio for unit volume

Tetrahedron


side

\sqrt{3} a^2

\frac{\sqrt{2}a^3}{12}

\frac{6\sqrt{6}}{a} \approx \frac{14.697}{a}

7.21

Cube


side

6a^2

a^3

\frac{6}{a}

6

Octahedron


side

2\sqrt{3}a^2

\frac{1}{3} \sqrt{2}a^3

\frac{3\sqrt{6}}{a} \approx \frac{7.348}{a}

5.72

Dodecahedron


side

3\sqrt{25+10\sqrt{5}} a^2

\frac{1}{4} (15+7\sqrt{5}) a^3

\frac{12\sqrt{25+10\sqrt{5}}}{(15+7\sqrt{5})a} \approx \frac{2.694}{a}

5.31

Icosahedron


side

5\sqrt{3}a^2

\frac{5}{12} (3+\sqrt5)a^3

\frac{12 \sqrt{3}}{(3+\sqrt{5})a} \approx \frac{3.970}{a}

5.148

Sphere


radius

4\pi a^2

\frac{4\pi a^3}{3}

\frac{3}{a}

4.836

Example Table
side of cube

side^{2}

Area of side

6*side^{2}

Area of Cube's Surface

side^{3}

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

See also
References

SchmidtNielsen, Knut (1984). Scaling: Why is Animal Size so Important?. New York, NY: Cambridge University Press.

Vogel, Steven (1988). Life's Devices: The Physical World of Animals and Plants. Princeton, NJ: Princeton University Press.
External links

Sizes of Organisms: The Surface Area:Volume Ratio

National Wildfire Coordinating Group: Surface Area to Volume Ratio

Previous link not working, references are in this document, PDF
Further reading

On Being the Right Size, J.B.S. Haldane
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