![Comparison of surface area vs volume of shapes](https://upload.wikimedia.org/wikipedia/commons/5/5f/Comparison_of_surface_area_vs_volume_of_shapes.svg)
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
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
per unit volume of an object or collection of objects.
SA:V is an important concept in science and engineering. It is used to explain the relation between structure and function in processes occurring through the surface ''and'' the volume. Good examples for such processes are processes governed by the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
, i.e., diffusion and heat transfer by conduction. SA:V is used to explain the diffusion of small molecules, like
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
and
carbon dioxide
Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
between air, blood and cells,
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
loss by animals, bacterial morphogenesis, organism's
thermoregulation
Thermoregulation is the ability of an organism to keep its body temperature within certain boundaries, even when the surrounding temperature is very different. A thermoconforming organism, by contrast, simply adopts the surrounding temperature ...
, design of artificial bone tissue, artificial lungs and many more biological and biotechnological structures. For more examples see Glazier.
The relation between SA:V and diffusion or heat conduction rate is explained from flux and surface perspective, focusing on the surface of a body as the place where diffusion, or heat conduction, takes place, i.e., the larger the SA:V there is more surface area per unit volume through which material can diffuse, therefore, the diffusion or heat conduction, will be faster. Similar explanation appears in the literature: "Small size implies a large ratio of surface area to volume, thereby helping to maximize the uptake of nutrients across the plasma membrane", and elsewhere.
For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
, a consequence of the
isoperimetric inequality in 3 dimensions. By contrast, objects with acute-angled spikes will have very large surface area for a given volume.
SA:V for balls and ''n''-balls
A
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
is a
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
object, being the solid version of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. (In geometry, the term ''sphere'' properly refers only to the surface, so a sphere thus lacks
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
in this context.) Balls exist in any dimension and are generically called ''n''-balls, where ''n'' is the number of dimensions.
![SAV n3](https://upload.wikimedia.org/wikipedia/commons/a/ac/SAV_n3.png)
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 has an inverse relationship with the radius - if the radius is doubled the SA:V halves (see figure).
The same reasoning can be generalized to n-balls using the general equations for volume and surface area, which are:
volume =
; surface area =
![SAV](https://upload.wikimedia.org/wikipedia/commons/8/84/SAV.png)
So the ratio reduces to
. Thus, the same linear relationship between area and volume holds for any number of dimensions (see figure): doubling the radius always halves the ratio.
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
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
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
:
.
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
shape.
Physical chemistry
Materials with high
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
to
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
ratio (e.g. very small diameter, very
porous
Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
, or otherwise not
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
) 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
![Human jejunum microvilli 2 - TEM](https://upload.wikimedia.org/wikipedia/commons/f/f2/Human_jejunum_microvilli_2_-_TEM.jpg)
The ratio between the surface area and volume of
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
and
organism
In biology, an organism () is any living system that functions as an individual entity. All organisms are composed of cells (cell theory). Organisms are classified by taxonomy into groups such as multicellular animals, plants, and ...
s has an enormous impact on their
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, including their
physiology
Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
and
behavior
Behavior (American English) or behaviour (British English) is the range of actions and mannerisms made by individuals, organisms, systems or artificial entities in some environment. These systems can include other systems or organisms as wel ...
. 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 feeder
Filter feeders are a sub-group of suspension feeding animals that feed by straining suspended matter and food particles from water, typically by passing the water over a specialized filtering structure. Some animals that use this method of feedin ...
s such as
krill
Krill are small crustaceans of the order Euphausiacea, and are found in all the world's oceans. The name "krill" comes from the Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in n ...
provide a large surface area to sift the water for food.
Individual organs like the
lung
The lungs are the primary organs of the respiratory system in humans and most other animals, including some snails and a small number of fish. In mammals and most other vertebrates, two lungs are located near the backbone on either side of t ...
have numerous internal branchings that increase the surface area; in the case of the lung, the large surface supports gas exchange, bringing
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
into the blood and releasing
carbon dioxide
Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
from the blood.
Similarly, the
small intestine
The small intestine or small bowel is an organ in the gastrointestinal tract where most of the absorption of nutrients from food takes place. It lies between the stomach and large intestine, and receives bile and pancreatic juice through the p ...
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
Microvilli (singular: microvillus) are microscopic cellular membrane protrusions that increase the surface area for diffusion and minimize any increase in volume, and are involved in a wide variety of functions, including absorption, secretion, ...
lining the
small intestine
The small intestine or small bowel is an organ in the gastrointestinal tract where most of the absorption of nutrients from food takes place. It lies between the stomach and large intestine, and receives bile and pancreatic juice through the p ...
.
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
biological rule
A biological rule or biological law is a generalized law, principle, or rule of thumb formulated to describe patterns observed in living organisms. Biological rules and laws are often developed as succinct, broadly applicable ways to explain comp ...
s such as
Allen's rule
Allen's rule is an ecogeographical rule formulated by Joel Asaph Allen in 1877, broadly stating that animals adapted to cold climates have thicker limbs and bodily appendages than animals adapted to warm climates. More specifically, it states that ...
,
Bergmann's rule
Bergmann's rule is an ecogeographical rule that states that within a broadly distributed taxonomic clade, populations and species of larger size are found in colder environments, while populations and species of smaller size are found in warmer ...
and
gigantothermy Gigantothermy (sometimes called ectothermic homeothermy or inertial homeothermy) is a phenomenon with significance in biology and paleontology, whereby large, bulky ectothermic animals are more easily able to maintain a constant, relatively high bod ...
.
Fire spread
In the context of
wildfire
A wildfire, forest fire, bushfire, wildland fire or rural fire is an unplanned, uncontrolled and unpredictable fire in an area of Combustibility and flammability, combustible vegetation. Depending on the type of vegetation present, a wildfire ...
s, 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.
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 (r=263 km), the ratio is so high that astronomers were surprised to find that it ''did'' differentiate and have brief volcanic activity. The
moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
,
Mercury
Mercury commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
and
Mars
Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury (planet), Mercury. In the English language, Mars is named for the Mars (mythology), Roman god of war. Mars is a terr ...
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
Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never fa ...
and
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
(r>6,000 km) have sufficiently low surface area-to-volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal.
Mathematical examples
See also
*
Compactness measure of a shape
The compactness measure of a shape is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space.
Properties
Various compactness measures are us ...
*
Dust explosion
A dust explosion is the rapid combustion of fine particles suspended in the air within an enclosed location. Dust explosions can occur where any dispersed powdered combustible material is present in high-enough concentrations in the atmosphere or ...
*
Square–cube law
The square–cube law (or cube–square law) is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the surface area as a shape's size increases or decreases. It was first ...
*
Specific surface area
Specific surface area (SSA) is a property of solids defined as the total surface area of a material per unit of mass, (with units of m2/kg or m2/g) or solid or bulk volume (units of m2/m3 or m−1).
It is a physical value that can be used to det ...
References
*
*
;Specific
External links
Sizes of Organisms: The Surface Area:Volume RatioNational Wildfire Coordinating Group: Surface Area to Volume RatioPrevious link not working, references are in this document, PDF
Further reading
{{Webarchive, url=https://web.archive.org/web/20110822151104/http://irl.cs.ucla.edu/papers/right-size.html , date=2011-08-22
Chemical kinetics
Cell biology
Physiology
Ratios