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.
For a given volume, the object with the smallest surface area 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. Balls exist in any dimension and are generically called n-balls, where n is the number of dimensions.
For an ordinary three-dimensional ball, the SA:V can be calculated using the standard equations for the surface and volume, which are, respectively, and. For the unit case in which r = 1 the SA:V is thus 3. The SA:V decreases exponentially as the radius increases - if the radius is doubled the SA:V halves.
The same reasoning can be generalized to n-balls using the general equations for volume and surface area, which are:
volume = ; surface area =
So the ratio reduces to. Thus, the same linear relationship between area and volume holds for any number of dimensions : doubling the radius always halves the ratio.

Dimension

The surface-area-to-volume ratio has physical dimension L−1 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
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 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

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.
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.
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. Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.
Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine.
Increased surface area can also lead to biological problems. More contact with the environment through the surface of a cell or an organ 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 biological rules such as Bergmann's rule and gigantothermy.

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 surface-area-to-volume ratio of the fuel. 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, 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. Venus and Earth have sufficiently low surface area-to-volume ratios so that their heat loss is minimal.

Mathematical examples

Side of
cube		Side2Area of a
single face		6 × side2Area of
entire cube
Side3VolumeRatio of
surface area
to volume
22x246x2x2242x2x283:1
44x4166x4x4964x4x4643:2
66x6366x6x62166x6x62163:3
88x8646x8x83848x8x85123:4
1212x121446x12x1286412x12x1217283:6
2020x204006x20x20240020x20x2080003:10
5050x5025006x50x501500050x50x501250003:25
10001000x100010000006x1000x100060000001000x1000x100010000000003:500