What is the relationship between the length of a side and the surface area to volume ratio of a cell?

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For the most part, life occurs on a very small scale. Life is based on cells, and cells (with a few exceptions like egg cells) are small. How small? A eukaryotic cell is typically about 30 micrometers in diameter. That’s 30 millionths of a meter. A prokaryotic cell (the type of cell found in bacteria) can be anywhere from 300 times smaller to five times smaller. Why are cells, the basic units of life, so small?

The answer lies in the relationship between a cell’s surface area and its volume. Surface area is the amount of surface an object has.

For a cube, the formula for area is (length of a side)2 x 6.
For a sphere, the formula for area is 4 Π r2

Volume is the amount of space inside something.

For a cube, the formula for volume is (length of a side)3.
For a sphere, the formula is 4/3 Π r3

The surface area to volume ratio is an object’s surface area divided by its volume. So, for a cube that’s one centimeter on a side,

the surface area is 6cm2 (1cm x 1cm x 6),
the volume is 1 cm3 (1cm x 1cm x 1cm), and
the surface area to volume ratio is 6 units of surface: 1 unit of volume

As you can see from the formulas, surface area is a square function (side 2 x 6), while volume is a cubic function (side)3. As a result, as the size of an object increases, its ratio of surface area to volume decreases. Conversely, as the size of an object decreases, its ratio of surface area to volume increases.

Figure 1

Study the table above, which shows the area, volume, and surface area: volume ratio for a variety of cubes. Note that in a cube that’s 0.1cm on a side the surface area: volume ratio is 60:1. This ratio falls to 0.6:1 in a cube that’s 10cm on a side.

This becomes even clearer in the chart below.

Figure 2

But how does this relate to the size of cells? Cells are constantly exchanging substances with their environment, and this exchange largely happens by the diffusion of materials through the cell membrane, the outer boundary of a cell. Cells need to be small so that they have enough surface for molecules to be able to diffuse in and out. On a much larger scale, you can see this in the picture below.

Figure 3

This picture is a still from the main demonstration shown in the Surface Area, Volume, and Life video. The cubes are made of agar, a seaweed extract. The agar contains a pH indicator, and the cubes are fuchsia (dark pink) on the inside because their initial pH is basic. When placed in vinegar (an acid), the vinegar diffuses into the cubes. As the vinegar diffuses in, the pH indicator changes from fuchsia to white.

This is what the cubes look like after six minutes. Note that 100% of the smallest cube’s volume has been reached by the vinegar, while only 19% of the largest video is reached by diffusion. The basic idea: small size results in a high surface area to volume ratio, which enhances diffusion. And that’s why cells are small.

Figure 4

The same principle explains a variety of biological phenomena. Why do elephants have big ears? Because elephants are huge, their bodies have a very low surface area to volume ratio. This makes it very difficult for heat to diffuse away from the elephant’s body. To compensate, elephants have evolved huge, flat ears. The ears, being flat, increase the elephant’s surface area, while barely increasing the volume. Blood in the ears can release heat into the environment.

A marine flatworm: no lungs, no heart.

Flattening out structures is an adaptation that also explains why flatworms can survive without any specialized system for distributing oxygen or carbon dioxide throughout their bodies. Because they’re flat, they have a very high surface area to volume ratio. This allows oxygen to diffuse from water directly to their body cells, and for carbon dioxide to diffuse from their body cells back out to the surrounding environment.

Figure 5: Whales have a relatively low surface area to volume ratio

Reducing the surface area to volume ratio can also be an adaptive strategy. Think about marine mammals. None of them are very small (the size of mice or even rats). The smallest marine mammals are otters, and it’s no coincidence that most marine mammals are relatively large (think of walruses, dolphins, manatees, etc.). One marine mammal, the blue whale, is the largest animal ever to have evolved. Why are marine mammals large? It’s instructive to compare marine mammals with elephants. Whereas elephants evolved huge ears as a way to enhance heat diffusion by increasing their surface area to volume ratio, marine mammals have evolved large sizes as a way of decreasing their surface area to volume ratio, as a way of decreasing heat loss. Look again at figure 3 above. The large cube, with its low surface area to volume ratio, has relatively little diffusion happening over time. And in terms of heat exchange, that’s advantageous for mammals living in cool ocean water. In other words, just by being large, marine mammals were able to evolve a strategy that enabled them to diminish heat loss.

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.

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. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.

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,[1] i.e., diffusion and heat transfer by conduction.[2] SA:V is used to explain the diffusion of small molecules, like oxygen and carbon dioxide between air, blood and cells,[3] water loss by animals,[4] bacterial morphogenesis,[5] organism's thermoregulation,[6] design of artificial bone tissue,[7] artificial lungs [8] and many more biological and biotechnological structures. For more examples see Glazier.[9]

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",[10] and elsewhere.[9][11][12]

For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a ball, 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.

A ball is a three-dimensional object, being the solid version of a sphere. (In geometry, the term sphere properly refers only to the surface, so a sphere thus lacks volume in this context.) Balls exist in any dimension and are generically called n-balls, where n is the number of dimensions.

Plot of the surface-area:volume ratio (SA:V) for a 3-dimensional ball, showing the ratio decline inversely as the radius of the ball increases.

For an ordinary three-dimensional ball, the SA:V can be calculated using the standard equations for the surface and volume, which are, respectively, $4 π r 2 {\displaystyle 4\pi {r^{2}}}$ and $( 4 / 3 ) π r 3 {\displaystyle (4/3)\pi {r^{3}}}$ . 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 = $r n π n / 2 Γ ( 1 + n / 2 ) {\displaystyle r^{n}\pi ^{n/2} \over \Gamma (1+n/2)}$ ; surface area = $n r n − 1 π n / 2 Γ ( 1 + n / 2 ) {\displaystyle nr^{n-1}\pi ^{n/2} \over \Gamma (1+{n/2})}$

Plot of surface-area:volume ratio (SA:V) for n-balls as a function of the number of dimensions and of radius size. Note the linear scaling as a function of dimensionality and the inverse scaling as a function of radius.

So the ratio reduces to $n r − 1 {\displaystyle nr^{-1}}$ . Thus, the same linear relationship between area and volume holds for any number of dimensions (see figure): doubling the radius always halves the ratio.

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 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

SA:V = 6   cm 2 1   cm 3 = 6   cm − 1 {\displaystyle {\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.

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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. 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.

Cells lining the small intestine increase the surface area over which they can absorb nutrients with a carpet of tuftlike microvilli.

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.[citation needed]

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.[13]

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.[14][15] Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.[16]

Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine.[17]

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.[citation needed]

The surface to volume ratios of organisms of different sizes also leads to some biological rules such as Allen's rule, Bergmann's rule[18][19][20] and gigantothermy.[21]

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.

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, 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.[22] Venus and Earth (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.[23]

Shape	Characteristic length $a {\displaystyle a}$	Surface area	Volume	SA/V ratio	SA/V ratio for unit volume
Tetrahedron	edge	$3 a 2 {\displaystyle {\sqrt {3}}a^{2}}$	$2 a 3 12 {\displaystyle {\frac {{\sqrt {2}}a^{3}}{12}}}$	$6 6 a ≈ 14.697 a {\displaystyle {\frac {6{\sqrt {6}}}{a}}\approx {\frac {14.697}{a}}}$	7.21
Cube	edge	$6 a 2 {\displaystyle 6a^{2}}$	$a 3 {\displaystyle a^{3}}$	$6 a {\displaystyle {\frac {6}{a}}}$	6
Octahedron	edge	$2 3 a 2 {\displaystyle 2{\sqrt {3}}a^{2}}$	$1 3 2 a 3 {\displaystyle {\frac {1}{3}}{\sqrt {2}}a^{3}}$	$3 6 a ≈ 7.348 a {\displaystyle {\frac {3{\sqrt {6}}}{a}}\approx {\frac {7.348}{a}}}$	5.72
Dodecahedron	edge	$3 25 + 10 5 a 2 {\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}a^{2}}$	$1 4 ( 15 + 7 5 ) a 3 {\displaystyle {\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}$	$12 25 + 10 5 ( 15 + 7 5 ) a ≈ 2.694 a {\displaystyle {\frac {12{\sqrt {25+10{\sqrt {5}}}}}{(15+7{\sqrt {5}})a}}\approx {\frac {2.694}{a}}}$	5.31
Capsule	radius (R)	$4 π a 2 + 2 π a ⋅ 2 a = 8 π a 2 {\displaystyle 4\pi a^{2}+2\pi a\cdot 2a=8\pi a^{2}}$	$4 π a 3 3 + π a 2 ⋅ 2 a = 10 π a 3 3 {\displaystyle {\frac {4\pi a^{3}}{3}}+\pi a^{2}\cdot 2a={\frac {10\pi a^{3}}{3}}}$	$12 5 a {\displaystyle {\frac {12}{5a}}}$	5.251
Icosahedron	edge	$5 3 a 2 {\displaystyle 5{\sqrt {3}}a^{2}}$	$5 12 ( 3 + 5 ) a 3 {\displaystyle {\frac {5}{12}}(3+{\sqrt {5}})a^{3}}$	$12 3 ( 3 + 5 ) a ≈ 3.970 a {\displaystyle {\frac {12{\sqrt {3}}}{(3+{\sqrt {5}})a}}\approx {\frac {3.970}{a}}}$	5.148
Sphere	radius	$4 π a 2 {\displaystyle 4\pi a^{2}}$	$4 π a 3 3 {\displaystyle {\frac {4\pi a^{3}}{3}}}$	$3 a {\displaystyle {\frac {3}{a}}}$	4.83598

Examples of cubes of different sizes

Side of cube	Side2	Area of a single face	6 × side2	Area of entire cube (6 faces)	Side3	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
50	50x50	2500	6x50x50	15000	50x50x50	125000	3:25
1000	1000x1000	1000000	6x1000x1000	6000000	1000x1000x1000	1000000000	3:500

Compactness measure of a shape
Dust explosion
Square–cube law
Specific surface area

Schmidt-Nielsen, Knut (1984). Scaling: Why is Animal Size so Important?. New York, NY: Cambridge University Press. ISBN 978-0-521-26657-4. OCLC 10697247.
Vogel, Steven (1988). Life's Devices: The Physical World of Animals and Plants. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08504-3. OCLC 18070616.

Specific

^ Planinšič, Gorazd; Vollmer, Michael (February 20, 2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics. 29 (2): 369–384. Bibcode:2008EJPh...29..369P. doi:10.1088/0143-0807/29/2/017. Retrieved 9 July 2021.
^ Planinšič, Gorazd (2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics European Physical Society, Find Out More. 29 (2): 369–384. Bibcode:2008EJPh...29..369P. doi:10.1088/0143-0807/29/2/017.
^ Williams, Peter; Warwick, Roger; Dyson, Mary; Bannister, Lawrence H. (2005). Gray's Anatomy (39 ed.). Churchill Livingstone. pp. 1278–1282.
^ Jeremy M., Howard; Hannah-Beth, Griffis; Westendorf, Rachel; Williams, Jason B. (2019). "The influence of size and abiotic factors on cutaneous water loss". Advances in Physiology Education. 44 (3): 387–393. doi:10.1152/advan.00152.2019. PMID 32628526.
^ Harris, Leigh K.; Theriot, Julie A. (2018). "Surface Area to Volume Ratio: A Natural Variable for Bacterial Morphogenesis". Trends in Microbiology. 26 (10): 815–832. doi:10.1016/j.tim.2018.04.008. PMC 6150810. PMID 29843923.
^ Louw, Gideon N. (1993). Physiological Animal Ecology. Longman Pub Group.
^ Nguyen, Thanh Danh; Olufemi E., Kadri; Vassilios I., Sikavitsas; Voronov, Roman S. (2019). "Scaffolds with a High Surface Area-to-Volume Ratio and Cultured Under Fast Flow Perfusion Result in Optimal O2 Delivery to the Cells in Artificial Bone Tissues". Applied Sciences. 9 (11): 2381. doi:10.3390/app9112381.
^ J. K, Lee; H. H., Kung; L. F., Mockros (2008). "Microchannel Technologies for Artificial Lungs: (1) Theory". ASAIO Journal. 54 (4): 372–382. doi:10.1097/MAT.0b013e31817ed9e1. PMID 18645354. S2CID 19505655.
^ a b Glazier, Douglas S. (2010). "A unifying explanation for diverse metabolic scaling in animals and plants". Biological Reviews. 85 (1): 111–138. doi:10.1111/j.1469-185X.2009.00095.x. PMID 19895606. S2CID 28572410.
^ Alberts, Bruce (2002). "The Diversity of Genomes and the Tree of Life". Molecular Biology of the Cell, 4th edition. New York: Garland Science. ISBN 0-8153-3218-1. ISBN 0-8153-4072-9.
^ Adam, John (2020-01-01). "What's Your Sphericity Index? Rationalizing Surface Area and Volume". Virginia Mathematics Teacher. 46 (2).
^ Okie, Jordan G. (March 2013). "General models for the spectra of surface area scaling strategies of cells and organisms: fractality, geometric dissimilitude, and internalization". The American Naturalist. 181 (3): 421–439. doi:10.1086/669150. ISSN 1537-5323. PMID 23448890. S2CID 23434720.
^ Kils, U.: Swimming and feeding of Antarctic Krill, Euphausia superba - some outstanding energetics and dynamics - some unique morphological details. In Berichte zur Polarforschung, Alfred Wegener Institute for Polar and Marine Research, Special Issue 4 (1983): "On the biology of Krill Euphausia superba", Proceedings of the Seminar and Report of Krill Ecology Group, Editor S. B. Schnack, 130-155 and title page image.
^ Tortora, Gerard J.; Anagnostakos, Nicholas P. (1987). Principles of anatomy and physiology (Fifth ed.). New York: Harper & Row, Publishers. pp. 556–582. ISBN 978-0-06-350729-6.
^ Williams, Peter L; Warwick, Roger; Dyson, Mary; Bannister, Lawrence H. (1989). Gray's Anatomy (Thirty-seventh ed.). Edinburgh: Churchill Livingstone. pp. 1278–1282. ISBN 0443-041776.
^ Romer, Alfred Sherwood; Parsons, Thomas S. (1977). The Vertebrate Body. Philadelphia, PA: Holt-Saunders International. pp. 349–353. ISBN 978-0-03-910284-5.
^ Krause J. William (July 2005). Krause's Essential Human Histology for Medical Students. Universal-Publishers. pp. 37–. ISBN 978-1-58112-468-2. Retrieved 25 November 2010.
^ Meiri, S.; Dayan, T. (2003-03-20). "On the validity of Bergmann's rule". Journal of Biogeography. 30 (3): 331–351. doi:10.1046/j.1365-2699.2003.00837.x.
^ Ashton, Kyle G.; Tracy, Mark C.; Queiroz, Alan de (October 2000). "Is Bergmann's Rule Valid for Mammals?". The American Naturalist. 156 (4): 390–415. doi:10.1086/303400. JSTOR 10.1086/303400. PMID 29592141. S2CID 205983729.
^ Millien, Virginie; Lyons, S. Kathleen; Olson, Link; et al. (May 23, 2006). "Ecotypic variation in the context of global climate change: Revisiting the rules". Ecology Letters. 9 (7): 853–869. doi:10.1111/j.1461-0248.2006.00928.x. PMID 16796576.
^ Fitzpatrick, Katie (2005). "Gigantothermy". Davidson College. Archived from the original on 2012-06-30. Retrieved 2011-12-21.
^ "Marsquake! NASA's InSight Lander Feels Its 1st Red Planet Tremor". Space.com. 23 April 2019.
^ "Archived copy" (PDF). Archived from the original (PDF) on 2018-06-13. Retrieved 2018-08-22.{{cite web}}: CS1 maint: archived copy as title (link)

Sizes of Organisms: The Surface Area:Volume Ratio Archived 2017-08-14 at the Wayback Machine
National Wildfire Coordinating Group: Surface Area to Volume Ratio
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