The “Square Cube Law”
In: Mathematics
You have a cube with side length a. Its surface area is 6×a² and its volume is a³. If I double the side length a to 2a the surface area charges by 6×a² -> 6×(2a)² = 6×4×a² and its volume changes to 8×a³. So given some scaling factor b if you scale the side length by b the surface area grows by b² and the volume by b³.
And this proportionality is generally true for any object. For a sphere we have the radius r and if we scale that by b we get for the surface area 4pi b² r² and for volume 4/3 pi b³ r³. (And if an object isn’t approximately sphere it’s approximately a cube.) But try different shapes the reactions that area scales by the square of some characteristic length and volume by its cube always holds.
Knowing that you can make the observation that if surface area is proportional to size² and volume to size³ if you scale something down its volume relative to its surface area decreases. The ratio of volume to surface area would be proportional to size³/size² = size so if you scale something down this ratio decreases linearly with the size meaning less volume per unit surface area. This is a useful fact if you are a cell or a bacteria trying to use diffusion for breathing, it happens through the surface of the microbe so it can’t get too big before its breathing surface is too small for all of its stuff in the bulk. Or say something is getting too hot like a computer chip, if you can scale it down you can cool it more efficiently as you can take away heat from the surface. The amount of heat generated would be proportional the the amount of stuff generating heat which is proportional to volume and you can cool from the surface. So as surface area increases relative to volume your cooling efficiency increases.
So basically the takeaway is that as you vary the size of an object its volume to surface area ratio is never constant. If you need less volume relative to surface area you need miniaturisation and if you need more volume relative to surface area you need to think big.
This law famously shows up in calculating terminal velocity for a falling object. You reach terminal velocity when weight (mg) equals air resistance (~A×v²), since air resistance is proportional to v² there is some velocity at with your constant mg equals that. If you have a specific shape and density in mind you can write m in terms of volume and density and now you have a force proportional to volume and the other proportional to surface area. If you divide by A you get the volume/area ratio and so the terminal velocity decreases as you scale the object down. Thats why ants take no fall damage.
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