Surface-area-to-volume ratio

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.

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. The surface-area-to-volume ratio is measured in units of inverse distance. As an example, a cube with sides of length a will have a surface area of 6a2 and a volume of a3. The surface to volume ratio for a cube is thus shown as $SA:V = \frac{6a^2}{a^3} = \frac{6}{a}$.

For a given shape, SA:V is inversely proportional to size. A cube 2 m on a side has a ratio of 3 m−1, half that of a cube 1 m on a side. Conversely, preserving SA:V as size increases requires changing to a less compact shape.

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.

Physical Chemistry

Materials with high surface area to volume ratio (e.g., very small diameter, or 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.

Surface to Volume Ratio causes a biological cell to be limited in size due to the fact that when the volume increases so does the surface area but the surface area increases at a slower rate.

Biology

The ratio between the surface area and volume of cells and organisms has an enormous impact on their biology (the physiology, behaviour, 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 low 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.

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.

Examples

ShapeCharacteristic Length $a$Surface AreaVolumeSA/V ratioSA/V ratio for unit volume
Tetrahedronside$\sqrt{3} a^2$$\frac{\sqrt{2}a^3}{12}$$\frac{6\sqrt{6}}{a} \approx \frac{14.697}{a}$7.21
Cubeside$6a^2$$a^3$$\frac{6}{a}$6
Octahedronside$2\sqrt{3}a^2$$\frac{1}{3} \sqrt{2}a^3$$\frac{3\sqrt{6}}{a} \approx \frac{7.348}{a}$5.72
Dodecahedronside$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
Icosahedronside$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
Sphereradius$4\pi a^2$$\frac{4\pi a^3}{3}$$\frac{3}{a}$4.836
Example Table
side of cubeside2Area of side6*side2Area of Cube's Surfaceside3VolumeRatio of Surface Area to Volume
22x246x2x2242x2x283:1
44x4166x4x4964x4x4643:2
66x6366x6x62166x6x62163:3
88x8646x8x83848x8x85123:4
1212x121446x12x1286412x12x1217283:6
2020x204006x20x20240020x20x2080003:10