There are very few things that are so far reaching across many different disciplines, ranging from biology to engineering, as is the relation of the surface area to the volume of a body. This is not a law, as Newton’s second one, or a theory as Darwin’s evolution theory. But it has consequences in a diverse set of situations. It explains why cells are the size they are, why some animals have a strange morphology, why flour explodes while wheat grains don’t and many other phenomena that we will explore in this article.
What Is SA:V?
All bodies have a volume, and a surface area. It is as simple as dividing the area of the body in question by its volume to obtain the ratio we are interested in. Consider for example a cube of 1 m in side. It has a volume of 1 m³ (1 cubic meter). At the same time each of its 6 faces has an area of 1 m² (1 square meter), and a total surface area of 6 square meters, therefore the surface area to volume ratio (or SA:V for short) is 6/1 = 6 m-1. This ratio varies with the body size, if you do the same calculation for a cube of 2 m to a side, you get a SA:V ratio of 3 m-1, and for a 10 m cube the value goes down to 0.6 m-1. It will tend to zero as the cube gets larger. The SA:V ratio also depends on the morphology of the body, for a given volume, the sphere is the object that has the smallest SA:V ratio.
Why the Surface Area to Volume Ratio Is Important
The surface of a body is important in many ways, because many reaction and transfer processes are directly proportional to surface area. Some examples are:
Heat transfer: Heat is transferred to and from a body primarily through its surface, but a large body has less surface available per unit volume (and mass if the body is homogeneous). In the figure below, the larger cube has 54 units of surface and 27 of volume, with a SA:V ratio of 2. The smaller one, has 6 units of surface and one unit of volume, with a SA:V ratio of 6. That means that the smaller cube can be heated or cooled 3 times as fast as the larger one.
Animals need to get rid of excess heat, but this becomes increasingly difficult for big species that have a small surface area compared to their body mass. So, special adaptations are needed in order to cool off a large animal. Such is the case with the elephant in the feature image. The big ears and the skin wrinkles provide the additional surface that is needed for cooling. Small animals have the opposite problem, they lose heat at a very fast rate, so they must eat large quantities of food in order to replace the energy that has been lost as heat.
Objects that are very small have a very large SA:V ratio and therefore they heat very quickly. This is the reason why even metals can “burn”, like the steel wool shown in the image or the powder metals used as fuel.
Another dramatic example is grain dust, that can be explosive. The Great Mill disaster is a sad example of that. In 1878, the Washburn A Mill exploded along with several adjacent flour mills, killing 18 workers and destroying the largest industrial building in Minneapolis.
Biology: The cells in every living thing need a set of substances to get into it in order to fuel the cell reactions. At the same time waste products need to be taken out. Cells rely on diffusion through its membrane in order to move substances in and out. As the cells grow, their SA:V ratio becomes smaller, and the membrane area is no longer sufficient to move substances at the necessary rate. Because of this, the SA:V ratio does impose an upper limit on the size of a cell. On a larger scale, several body structures have evolved to maximize the SA:V ratio, such as the lungs and the intestines. Your intestines have no less than 300 m2 of surface area available for digestion.
Engineering: In engineering, the SA:V ratio is also known as the square-cube law. When an object is scaled up by some multiplier, its mass increases as the cube of the multiplier, but its surface and cross sectional area are increased only as the square of the multiplier.
Consider a very large airplane such as the Airbus 380. Its wings are proportionately larger that those of a smaller airplane such as the Boeing 747. Just scaling up the 747 to the size of the 380 will result in wings with not enough surface area to give enough lift for the weight of the airplane. A skycraper that is double the size of another in its dimensions has eight times the weight, but only four times the base area, therefore you may need different construction techniques, such as using steel rather than only wood and brick.
You may have noticed how different very small animals are compared to larger ones. Ants have legs that are very skinny compared to their bodies, and they can lift many times its own weight. Elephants on the other side, have thick legs and they cannot lift great weights compared to their own mass. The reason is the same, when the dimensions grow, the weight increases faster than the area of the legs, and the pressure exerted is significantly larger.
We have given just a few examples of the many instances where the SA:V ratio has an influence. It is hard to think of a discipline of human knowledge where this principle does not apply. In the end, we can say that yes, size does matter.