Today, we’re going to start our biology lesson with a little bit of geometry. The square-cube law is one of those deceptively simple fundamentals that can be easy to miss but becomes almost obvious in hindsight. According to Wikipedia, it can be described thus:
When an object undergoes a proportional increase in size, its new surface area is proportional to the square of the multiplier and its new volume is proportional to the cube of the multiplier.
-Wikipedia, Square-Cube Law
And that’s not just some guys on a wiki talking, either. There are several ways to describe the square-cube law mathematically, but the easiest might just come from looking at its namesake.
Here’s the Math
Basically, think of a square. The area of that square, A, comes from taking the length of the side s and multiplying it by itself. This gives you the formula A=s2.
But then if you look at a cube, that adds another dimension that’s still the same side length s. To get the volume of that cube, you have to multiply s by itself once to cover the height and width and then again to get the depth. Using V as our volume, this gets you V=s3.
So far, so standard geometry, right? But what we’re interested in is what happens when you increase the size of that cube. Let’s say you want to double the size of your cube, which you can do by doubling the size of all the sides. (Remember that the key word here is “proportional,” which means all the sides double at once and we’re still looking at a square and a cube).
Let’s start with a simple side length of 1. If we plug that into our area equation, we get A=s2=12=1. But when we double that to get a side length of two, it becomes A=s2=22=4.
Looking at the volume equation, we can do the same basic thing. Starting with a side length of 1, we get V=s3=13=1. But then when we double that for a side length of two, the equation becomes V=s3=23=8.
So, we started with both an area and a volume of 1. But then when we started to increase the side length, the volume increased by much more than the area. And this trend continues. Using a side length of 3, for example, you’ll get an area of 9 and a volume of 21.
(You might notice that the Wikipedia quote specifically mentions surface area, which is a little different from the area of a square. The math behind that is a little more complicated, but it still uses an exponent of 2 in the equation and the trend that results is largely the same. We won’t get into that today because our topic of interest only needs regular areas.)
So, what does this have to do with biology?
Quite a lot, actually! Because living things are three-dimensional, the square-cube law applies to them too. And that has an effect on everything from relative strength to practical insect size to how easy it is to fly.
That last one, of course, is what we’re interested in.
See, flight is all about competing forces. You have lift, thrust, drag, and gravity. Lift is upward motion and thrust is forward, which is nice. Drag is the wind pulling you back and gravity pulls you down, which makes things harder.
And the force gravity has on something increases with mass, which increases with volume. While lift comes from the wings, and is mostly concerned with their area.
See the problem?
Luckily, most things are not in fact literal squares and cubes. Animals can develop larger, wider wings relative to the size of their main bodies as needed. And they can offset the issue of mass by making themselves less dense, which is why most birds have what we call “hollow” bones.
So the moral of the story is: If you want to make a big flying beasty, give it really big wings. And maybe some funky inner anatomy.
Simple, right? Well… that’s only part of the story, unfortunately. We still have thrust and drag to worry about, so you can’t just slap some giant wings on your dragon, roc, or cockatrice and call it a day.
Luckily, there are answers. Large pterosaurs and teratorns, the biggest flying reptiles and birds respectively, managed it. They just had to get creative with their size, shape, and take-off. We’ll get into those details in the next post, but for now, just remember that the square-cube law is everywhere.
Everywhere. Our world is three-dimensional so far as we can tell, after all.
One response to “The Square-Cube Law and Giant Fliers: Part One”
[…] the last post, we learned a little bit about the square-cube law and how it influences flight mechanics. To recap, the bigger animals get, the harder it is to fly […]