A / V ratio

The surface-to-volume ratio ( A / V ratio ) is the quotient of the surface and the volume of a geometric body. It has the dimension 1 / length. ${\displaystyle A}$ ${\displaystyle V}$

A / V ratio of sphere, cube and cuboid (aspect ratio 1: 2: 3) with the same volume

For a given volume, the sphere has the smallest surface area of all bodies . As the volume increases, the A / V ratio decreases for all bodies, since the surface area increases as a square, but the volume increases in a cubic manner (to the power of three). This is important for the cooling rate of masses of different sizes: The cooling takes place in proportion to the size of the surface, which, however, grows more slowly than the volume as it increases, so that larger masses cool down more slowly than small ones. This also explains why emperor penguins in Antarctica are larger and therefore retain more heat than Galápagos penguins near the equator, which are more likely to want to give off heat ( Bergmann's rule and allometry ).

Consideration

In general, the following applies to bodies : if you double the edge length of a cuboid , its area quadruples (in general terms: surface ; or, if exchange processes are taken into account, its interface ); its volume but ver eight fanned herself. Large bodies therefore have a more favorable ratio of volume to surface (e.g. for heat storage) :

${\displaystyle V_{1}=a\times b\times c}$(the cube : ) ${\displaystyle V_{1}=a\times a\times a}$

${\displaystyle A_{1}=2\times a\times b+2\times b\times c+2\times a\times c}$(Cube: ) ${\displaystyle A=6\times a\times a}$

${\displaystyle V_{2}=2a\times 2b\times 2c=2\times 2\times 2\times a\times b\times c=8\times a\times b\times c}$

This also applies to the cylinder : if you double its diameter and height, its volume increases eightfold. Even if you double the diameter of a sphere , its volume increases eightfold. A sphere has the largest ratio of volume to surface of all geometric bodies .

Physiological implications

The exchange of substances in a cell takes place via its surface . The uptake and release of molecules important for the metabolism takes place via the cell membrane ( phase interfaces ). The ratio of cell surface to cell volume also plays an important role. The smaller a cell (or a body), the less volume it has in relation to its surface area. A metabolically active cell is therefore usually small, since the surface-to-volume ratio is more favorable with a small cell body than with large-volume cells. However, if a cell is to be both large in volume and metabolically active due to the evolutionary pressure , this is only possible through an additional increase in the surface area through folds or protuberances, an example being the osteoclast .

In organisms of different sizes, the surface-to-volume ratio leads to ecogeographical observations such as Bergmann's rule .

Building physics

In building physics and in proof of thermal insulation , the A / V ratio is an important parameter for the compactness of a building . It is calculated as the quotient of the heat-transferring envelope surface, i.e. H. Surfaces that give off heat to the environment, such as walls, windows, roofs and the heated building volume. The A / V ratio has a decisive influence on the heating energy requirement . A lower A / V ratio means a smaller heat-transferring external surface for the same building volume. This means that less energy is required per m³ of volume to compensate for the heat losses through the envelope.

Large buildings naturally have smaller A / V ratios than z. B. Single-family homes . Typical values ​​for single-family houses are between 0.8 and 1.0 . For large, compact buildings, values ​​below 0.2 are possible. ${\textstyle \mathrm {\frac {1}{m}} }$ ${\textstyle \mathrm {\frac {1}{m}} }$

Examples

body	shape	length ${\displaystyle a}$	surface	volume	A / V ratio	A / V ratio per unit of space
Tetrahedron		page	${\displaystyle {\sqrt {3}}a^{2}}$	${\displaystyle {\frac {{\sqrt {2}}a^{3}}{12}}}$	${\displaystyle {\frac {6{\sqrt {6}}}{a}}\approx {\frac {14{,}697}{a}}}$	7.21
cube		page	${\displaystyle 6a^{2}}$	${\displaystyle a^{3}}$	${\displaystyle {\frac {6}{a}}}$	6th
octahedron		page	${\displaystyle 2{\sqrt {3}}a^{2}}$	${\displaystyle {\frac {1}{3}}{\sqrt {2}}a^{3}}$	${\displaystyle {\frac {3{\sqrt {6}}}{a}}\approx {\frac {7{,}348}{a}}}$	5.72
Dodecahedron		page	${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}a^{2}}$	${\displaystyle {\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}$	${\displaystyle {\frac {12{\sqrt {25+10{\sqrt {5}}}}}{(15+7{\sqrt {5}})a}}\approx {\frac {2{,}694}{a}}}$	5.31
Icosahedron		page	${\displaystyle 5{\sqrt {3}}a^{2}}$	${\displaystyle {\frac {5}{12}}(3+{\sqrt {5}})a^{3}}$	${\displaystyle {\frac {12{\sqrt {3}}}{(3+{\sqrt {5}})a}}\approx {\frac {3{,}970}{a}}}$	5.148
Bullet		radius	${\displaystyle 4\pi r^{2}}$	${\displaystyle {\frac {4\pi r^{3}}{3}}}$	${\displaystyle {\frac {3}{r}}}$	4.836

Individual evidence

1. ↑ Hans Joachim Schlichting; Bernd Rodewald: About large and small animals. In: Praxis der Naturwissenschaften- Physik. 37/5, 2 (1988).
2. ↑ Werner Buselmaier: Biology for physicians. 12th edition. Springer, 2012, ISBN 978-3-642-27174-8 , p. 4 f.