The square-cube law is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. It was first demonstrated in 1638 in Galileo's Two New Sciences. It states:

When an object undergoes a proportional increase in size, its new volume is proportional to the cube of the multiplier and its new surface area is proportional to the square of the multiplier.

$v\_2=v\_1\backslash left(\backslash frac\{\backslash ell\_2\}\{\backslash ell\_1\}\backslash right)^3$

where $v\_1$ is the original volume, $v\_2$ is the new volume, $\backslash ell\_1$ is the original length and $\backslash ell\_2$ is the new length. Note that it doesn't matter which length is used.

$A\_2=A\_1\backslash left(\backslash frac\{\backslash ell\_2\}\{\backslash ell\_1\}\backslash right)^2$

where $A\_1$ is the original surface area and $A\_2$ is the new surface area.

For example, if a cube with a side length of 1 metre were doubled in size, its volume would be 8 m³ and its surface area would be 24 m². This principle apples to all solids.

The ratio of the volume to the surface area of one face is equal to the scaling factor.

Applications

Engineering

When a physical object maintains the same density and is scaled up its mass is increased by the cube of the multiplier while its surface area only increases by the square of said multiplier. This would mean that when the larger version of the object is accelerated at the same rate as the original, more pressure would be exerted on the surface of the larger object. The object would therefore 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 limits to how high buildings could be built. Similarly, the larger an object is, the less other objects would resist its motion, causing its deceleration.

Biomechanics

If an animal were scaled up by a considerable amount, its muscular strength would be severely reduced since the cross section of its muscles would increase by the square of the scaling factor while their mass would increase by the cube of the scaling factor. As a result of this, cardiovascular functions would be severely limited. In the case of flying animals, their wing loading would be increased if they were scaled up, and they would therefore have to fly faster to gain the same amount of lift. This would be difficult considering that muscular strength was reduced. This also explains how a bumblebee can have a large body relative to its wings, which would not be possible for a larger flying animal. Air resistance per unit mass is also higher for smaller animals, which is why a small animal like an ant cannot die by falling from any height (the exoskeleton helps, but a tank could no more survive a fall from a mile than an elephant could).

See also

• allometric law

References

AreaVolume