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.

When applied to the real world, this principle has many implications which are important in fields ranging from mechanical engineering to biomechanics.

It helps explain phenomena including why large mammals like elephants have a harder time cooling themselves than small ones like mice, and why building taller and taller skyscrapers is increasingly difficult.

For example, a cube with a side length of 1 meter has a surface area of 6 m2 and a volume of 1 m3.

The original cube (1 m sides) has a surface area to volume ratio of 6:1.

The larger (2 m sides) cube has a surface area to volume ratio of (24/8) 3:1.

As the dimensions increase, the volume will continue to grow faster than the surface area.

[3] When a physical object maintains the same density and is scaled up, its volume and mass are increased by the cube of the multiplier while its surface area increases only by the square of the same multiplier.

This would indicate that the object would have less ability to resist stress and would be more prone to collapse while accelerating.

This is why large vehicles perform poorly in crash tests and why there are theorized limits as to how high buildings can be built.

If an animal were isometrically scaled up by a considerable amount, its relative muscular strength would be severely reduced, since the cross-section of its muscles would increase by the square of the scaling factor while its mass would increase by the cube of the scaling factor.

As a result of this, cardiovascular and respiratory functions would be severely burdened.

In the case of flying animals, the wing loading would be increased if they were isometrically scaled up, and they would therefore have to fly faster to gain the same amount of lift.

Air resistance per unit mass is also higher for smaller animals (reducing terminal velocity) which is why a small animal like an ant cannot be seriously injured from impact with the ground after being dropped from any height.

This is due to allometric scaling: the bones of an elephant are necessarily proportionately much larger than the bones of a mouse because they must carry proportionately higher weight.

Haldane illustrates this in his seminal 1928 essay On Being the Right Size in referring to allegorical giants: "...consider a man 60 feet high...Giant Pope and Giant Pagan in the illustrated Pilgrim's Progress: ...These monsters...weighed 1000 times as much as [a normal human].

Every square inch of a giant bone had to support 10 times the weight borne by a square inch of human bone.

As the average human thigh-bone breaks under about 10 times the human weight, Pope and Pagan would have broken their thighs every time they took a step.

The giant creatures seen in monster movies (e.g., Godzilla, King Kong, and Them!, and other kaiju) are also unrealistic, given that their sheer size would force them to collapse.

Robert Wadlow, the documented tallest man to ever live (2.72m), needed leg braces to walk and suffered from numbness in his feet.

[6] However, the buoyancy of water negates to some extent the effects of gravity.

The metabolic rate of animals scales with a mathematical principle named quarter-power scaling[7] according to the metabolic theory of ecology.

Mass transfer, such as diffusion to smaller objects such as living cells is faster than diffusion to larger objects such as entire animals.

Thus, in chemical processes that take place on a surface – rather than in the bulk – finer-divided material is more active.

For example, the activity of a heterogeneous catalyst is higher when it is divided into finer particles.

Heat production from a chemical process scales with the cube of the linear dimension (height, width) of the vessel, but the vessel surface area scales with only the square of the linear dimension.

Also, large-scale piping for transferring hot fluids is difficult to simulate on a small scale, because heat is transferred faster out from smaller pipes.

Failure to take this into account in process design may lead to catastrophic thermal runaway.