Square-Cube Law and Human Height 3


The words tall and short refer to the vertical extent of something. So we call a person with a greater vertical extent tall and lesser short. But this can be misleading, as there are much greater differences than merely vertical extent between short and tall people. In fact, the fundamental properties of the body vary with height. The square-cube law can explain why.

Take a look at the smaller box in in the figure below. If you double its height while maintaining proportions, length and width also double. Less obvious though is that the volume increases more than the area, twice as much to be exact. This is because 3D volume is proportional to the cube of height while 2D area is merely proportional to the square of height. This is the square-cube law. Galeleo might have been the first to formally recognize this when he stated in his 1638 book, Two New Science, “…the ratio of two volumes is greater than the ratio of their surfaces.”

Square-Cube Law

This is enormously important as so much depends on the ratio of volume to area. For example, weight is proportional to volume while strength is proportional to area. So while strength increases with size, in relation to weight it actually decreases. Another way to say this is that the larger body is absolutely stronger but relatively weaker.

The classic example of this is a comparison between an elephant and an ant. The elephant is stronger, this is most obvious. Where the ant excels, though, is in being able to carry a great many times its own body weight. You see this when ants are carrying food many times their own size back to their queen. Elephants clearly can’t do this. The elephant is absolutely stronger but the ant is relatively stronger. This trend with size is present not just for ants and elephants, but all animals, even humans.

Square-Cube Law Ant Elephant

So while tall humans have greater absolute strength, they have lesser relative strength. This is why tall people, like world record holding 6’6″ Behdad Salimi Kordasiabi, can lift more weight but then aren’t as good at push-ups and chin-ups1. Similarly, this can explain why tall people, like 6’5″ Usaine Bolt, may be fast yet make for poor gymnasts due to a lesser ability to accelerate their own bodies.

Beyond forces, the square-cube law has implications for many other aspects of the body, such as thermal properties. Heat generated and contained is a function of volume while heat dissipated is a function of surface area. This means tall people should be better at maintaining body temperature in a cold climate though struggle to cool off in a hot climate or during exertion. This heat transfer relationship also means that taller bodies lose relatively less energy to the external environment, which in turn coincides with a lower metabolic rate in relation to mass. In other words, the taller body is less energy efficient in the absolute sense, yet more so in the relative sense. This has implications for longevity.

This article has assumed that people maintain proportions as they scale. In actuality, however, taller people tend to be relatively skinnier. Furthermore, some tall people arrive at their height via long limbs. These factors can change the relationships discussed above.

  1. Sekerak RJ, Zimmermann KP. Chin-up strength tests: does stature matter? J Sports Med Phys Fitness. 2008;48(1).

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