The Square-Cube Law

By Earl Hunsinger

You probably remember watching B movies when you were a kid. You know the type—the kind where giant ants take over the world or some other equally terrorizing, but implausible, thing happens. If you still lay awake at night worrying that they might come true, put your mind at ease, it will never happen. (At least the one about the ants won’t; for the others, you’re on your own).

Ants, and other things, are subject to a scientific principle called the square-cube law, which limits their size. This is how it works (bear with me, I promise it will get more interesting in a minute). For the sake of simplicity, consider a concrete cube one inch on a side—in other words, one inch wide, one inch tall, and one inch deep. The surface area of this cube is six square inches. The volume is one cubic inch. The cross-sectional area is one square inch. This is important because it is this cross sectional area that has to support the weight of the cube. The weight is proportional to the volume, with one cubic inch of concrete weighing a certain amount. Of course, concrete is very strong, so it’s no problem for the one square inch bottom of the cube to support the weight of one cubic inch of concrete.

What happens, though, if we double the size of the cube in each of its dimensions? In other words, let’s make it two square inches on a side. This means that the cross sectional area, or the bottom of the cube, is equal to four square inches. The volume is calculated by multiplying two inches by two inches by two inches. So the volume is eight cubic inches. This means that even though the bottom of the cube is only four times bigger, the weight of the concrete cube is eight times greater, and each square inch of the cross sectional area has to support the weight, not of one cubic inch of concrete, but of two. Since concrete is very strong, this is still no problem.

However, the same thing happens each time we increase the size of the cube. If we make it a thousand times bigger on each side, the bottom will be a million times bigger (1,000 x 1,000) but the volume, and therefore the weight, will be a billion times greater (1,000 x 1,000 x 1,000). Now each square inch of bottom will have to support a thousand cubic inches of concrete. If we keep increasing its size in this way, eventually it will collapse under its own weight (this of course assumes that the ground under it has been capable of supporting it until it reaches this point). This is one of the factors limiting how tall a building can be built.

Finally, we are brought back to ants. Instead of concrete, consider that our cube represents a chunk of an ant’s leg. Of course an ant’s leg is not square and it’s much smaller than an inch across, but the same principle can be applied to the actual size and shape of an ant’s leg. The only difference is that the math is harder (you can do the math if you want to, but I’m not going to).

Ants are incredibly strong, being able to lift many times their own weight and survive falls from tremendous heights in proportion to their size. This is especially impressive when you see how skinny an ant’s legs are compared to its body. However, if you were able to somehow enlarge an ant (in the movies, radiation seems to be a favorite method of enlarging things), the cross sectional area of our imaginary block of ant leg would increase at a slower rate than its volume, and therefore its weight.

This is important for the same reason that it was with our concrete block, because it is this cross-sectional area that has to support the weight of the block of ant leg, and of course the rest of the leg and the enlarging ant above the leg. If the leg becomes ten times thicker, its cross-sectional area becomes something like 100 times larger. At the same time though, its weight becomes something like 1,000 times greater. If you keep doing this, at some point, long before the ant is large enough to take over even a small town, much less the world, it will collapse under its own weight, just like the concrete cube.

For other problems associated with enlarging ants, check out the article on Size and Scaling from Dr. Thomas J. Herbert, Professor of Biology at the University of Miami.

The square-cube law explains why insects have proportionally thinner legs than larger creatures, and yet can jump off tables, or out of airplanes, and not get hurt. It also explains why elephants, which have incredibly thick legs in proportion to their body size, do not climb trees (some might argue that this is only one of the reasons).

This scientific principle can also be applied in the other direction, to explain why people cannot be shrunk down to the size of ants (again, there might be other factors involved here). But perhaps this discussion is best left to another time. Until then, suffice it to say that you should be able to sleep better at night now, confident that there is no chance that giant ants will take over the world while you’re asleep.