Can anyone explain in simple terms the square cube law and how it applies to animals and people. I have seen it referenced to fictional beast, especially those who are absurdly large. But can anyone explain it what it actually means
In: Biology
As surface area grows; volume grows much larger
Skynscrapers have to become thinner the higher up they go to prevent structural failure
Elephants and rhinos struggle to stay cool relative to mice and squirrels
If you have a cube filled with water; and make the cube larger (and still filled with water) the pressure on the sides will be significantly higher…..to solve this you’d need thicker walls; or lower pressure by removing some water
The square-cube law refers to how different properties of an object or creature change in different proportions (by the difference squared, or the difference cubed).
One example of this is weight. Your weight is a function of your mass, which is indirectly tied to your volume; the bigger your three-dimensional size (three dimensions is analogous to a cubic function of length), the more mass you have and the heavier you weigh.
In order to support your weight, you need legs and either bones or a shell. The strength of muscles/bones/shell is proportional to their thickness, which is usually calculated as the cross-sectional area of the (think of the “I” shape of an I-beam). A longer bone doesn’t carry more weight, only a thicker/wider bone does. An area is analogous is a square function of length; thus, the larger an animal gets, the bigger its bones can be, and the more weight it can carry.
However, the rate of increase for these two attributes is not equal; one increases by the square and the other by the cube. If the animal gets too large, the weight will be too much for the bones to carry, and it will not be able to move around or support itself at all.
Another example of this is heat. Each one of your cells creates heat just by being alive. The only way to get rid of this heat and not cook yourself is by emitting it the environment through your skin or breath. However, the amount of skin you have increases in a square fashion, while your body mass increases in a cubic fashion. You can cheat a bit by having wrinkly skin, but that only helps a little because the air around you doesn’t transfer heat that well. As you increase in size, you generate a lot more heat, but can only get rid of a little more heat. At some point, the rates will no longer match causing your body to heat up, like a permanent fever. If the fever gets too high, your body simply won’t be able to operate and you’ll die.
It just means that if you increase the dimensions of something, the surface area grows with the square of the size and the volume grows with the cube if the size. For example, if you double the size of a cube, the length of its sides increase 2x, it’s surface area increases 4x and it’s volume increases 8x.
This has implications in biology because it puts limits on how large organisms can get. For example, if you double the size of an animal, it’s mass increases 8x, but the cross-section of it’s muscle, and therefore it’s strength, only increases 4x. If it existed, it would collapse under it’s own weight and die.
Other people have explained the square cube law just fine, but in terms of biology, the mass of an animal affects how much it has to eat, how it breathes, what ecological niche it fills, and sometimes how it reproduces.
Take it down to the tiny level then. Think of the cells in your body. Let’s assume they’re spherical to make life easier. Volume of a sphere = 4/3 * pi * radius^3 and surface area = 4 * pi * radius^2
Therefore, as the cell increases in size, its volume is rising much faster than its surface area. Obviously a bigger cell needs to carry out more processes than a smaller one (metabolism etc.) so it generates a lot more metabolic waste than a small cell, but doesn’t have the corresponding increase in surface area to get things in and out of the cell.
This is why when a cell reaches a certain size it becomes advantageous for it to split into two identical cells (mitosis) thus improving the surface area to volume ratio.
It’s used in two contexts.
One is literal to the namesake: as an object grows, it’s surface area grows as a square of diameter, while volume grows as a cube. This is important for heat generation and retention of organisms, as the heat is generated by the bulk (volume) of the body, but absorbed or expelled by the skin (surface). This means that as an animal grows larger, it retains heat better, but conversely also has trouble shedding it.
In other words, mice freeze really easily, while elephants overheat (if not for evolved adaptations).
The other one is a bit more tangential, but relevant to ***everything***. The volume increases with the cube of diameter. And that means mass does too. This means the mass of an object increases far faster than its dimensions would suggest, and that also means it can quickly outgrow the strength of the materials it’s made of. That’s why you can make a tiny toy bridge out of playdoh, but won’t even be able to finish building a life sized one without it collapsing on itself.
The same goes for animals, their bones, sinews etc. are stronger compared to the body weight for a small animal than a big one. That’s why a ferret can fall on its head from many times it’s body length and be fine, but a human falling on their head from even literally the same height (not to mention one proportionally scaled) will get injured.
For particularly large animals, say elephants or dinosaurs, their bones have to grow ridiculously thick compared to their body size to not snap under their body weight. And for particularly ridiculous fictional creatures, they just simply cannot be made from real life flesh and bone, because there’s no way they wouldn’t collapse on themselves.
Let’s simplify with a cube.
A cube has 6 square sides, so it’s surface area is 6*s*^(2), where *s* is the length of a side.
A volume is just *s*^3
So the surface area depends on the square of *s*, whereas the volume depends on the cube of *s*. Hence the name of the law.
This means volume gets bigger faster than the surface area. A 1x1x1 cube has a surface area of 6 and a volume of 1. A 100x100x100 cube has a surface area of 60,000 and a volume of *1,000,000*.
For living things, surface area and volume have different functions and effects.
For example, every cell in our body generates heat, but heat can only escape from our skin. Aka our volume creates heat, but our surface area removes heat. If we simply scale up a human, they’d likely just overheat because they can’t get rid of heat fast enough.
Another example is falling. More volume means more weight, obviously. But air resistance is based on surface area. That means bigger animals have a faster terminal velocity. An ant can survive any fall, but we’d get splattered from high falls.
It means “when things get bigger, their weight increases *super* quickly, much faster than their size does”.
A 10x10x10″ cube’s volume is 1000 cubic inches. If we double it to a 20x20x20″ cube, its volume is now 8000 cubic inches. A *2x* increase became an *8x* increase in volume (and therefore also an 8x increase in weight).
In biology, this means “animals can only be so large, because if you make them bigger they’ll be crushed under their own weight”. Ocean animals can “cheat” a bit because the water helps support their weight.
Have you ever heard of that funny trick of how a pizza twice as wide actually has four times as much pizza? So a 20″ pizza actually has four times as much pizza as a 10″ pizza. The square cube law is kind of like that.
It’s not so much a biological property as a mathematical one. Let’s take it from 2D (pizza) to 3D (sphere, or pizza’s final form). A sphere has exactly one linear property, radius, which is the difference from the center to every point on the surface of the sphere. The surface area is 4πr^2 while the volume is (4/3)πr^3 . The proofs of those formulae are not hard, but not ELI5. Just know that if the radius doubles, the surface area quadruples, and when the surface area quadruples, the volume is multiplied by 8 times.
So if you’ve got a sphere of radius 10 (the unit doesn’t matter). It has a radius of 10, a surface area of 1256.6, and a volume of 4188.8. If you double that radius, like we doubled the radius of our pizza, you have a sphere with radius 20, a surface area of 5026.5, and a volume of 33510. You see that the surface area increases much faster than radius, and volume increases much faster than surface area. That’s the square-cube relation.
The biological implication of this is outside of my area, but think about it in a simple sense: We human beings cool ourselves with evaporative cooling by sweat; if we double in linear size (height) while keeping all other dimensions proportionate, we would quadruple in surface area, but increase our internal volume by 8 times (give or take), meaning that we would, proportionately, have much less skin to sweat and cool us. We would all die of heat exhaustion. Similarly, an insect that doubled its size would have 4 times as much exoskeleton which means, if its bug-equivalent of muscles didn’t get much stronger, it wouldn’t be able to move.
Again, the biology is not my area, just the math, but if a biologist wanted to chime in and explain some of those biological implications better it would be awesome. But TL;DR – the square-cube relation means that things that increase in one dimension don’t increase in other dimensions the same way, so the same physical adaptations might not work the same.
EDIT – Typos and grammar
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