Imagine you have a cube of something. It has side length one, and six faces of area one (total surface area 6), and volume of one.

Now, if you want to double the side length, you’ll have a 2x2x2 cube. Six faces of area 4 (total 24), and the volume is now 8. So the volume has gone up by the scale factor cubed (faster), and the surface area has gone up by the scale factor squared (slower than cubed).

Every type of shape scales this way. So when you make something bigger, you have to worry about how surface area-dependent properties interact with volume-dependent properties.

In living systems, volume affects things like weight and how many cells you need and how much blood you need to supply it. Surface area affects things like how much oxygen you can absorb through your lungs and how much heat you can exchange with the environment. Cross sectional area (also scales like area) affects things like how much weight your bones and muscles can support.

If any of the square things can’t keep up with the cube things, you have a scaling problem.

As a shape grows in size, its surface area goes up as a mathematical square (power of 2) but its volume goes up as a cube (power of 3). This means that mass increases rapidly if it’s made of the same stuff and that increases pressure on the outside of the shape.

This has a lot of impacts on engineering, and also makes most classic “giant monster attack” stories physically impossible.

The square-cube law is the mathematical relationship between “area” (or “surface area”) and “volume” and is used to explain certain phenomena we observe.

Consider a living cell as a sphere with a radius of r = 1. The surface area of this sphere is C * r^2 = C, where “C” is some constant number, and the volume of this sphere is D * r^3 = D. where “D” is a different constant number. If you increase the radius to 2, the surface area becomes 4C, and the volume becomes 8D. At r = 3, we have 9C and 27D.

The point here is that, since volume is the radius *cubed*, but surface area is the radius *squared*, volume will grow much faster than surface area as the radius increases.

Why does this matter? Let’s go back to our cell. The cell has stuff inside it that it needs to fuel, and it had stuff outside it that it needs to gather for that fuel. But the only place that things can move from “outside” to “inside” is on the surface.

Let’s suppose the cell eats bugs. With a radius of 1, it has a volume of ~1, so it needs 1 bug per hour to fuel itself. It has a surface area of ~1, so it can move 1 bug per hour through its surface. One bug in, one bug used, no problem.

At r = 2, we can move 4 bugs per hour through the surface, but we need 9 bugs per hour to fuel the inside! At r = 3, the cell can move 9 bugs per hour but needs to eat 27.

The punchline here is that this law makes it harder for living things, or really any similar systems, from getting too big.

Take some legos and build a tiny cube, let’s say four of the four by two bricks.

Now you want to make another cube that is twice as big in every direction.

To make it twice as long you need to double the amount of bricks you use, so 8.

To make it twice as wide you need to double the amount again, so 16.

To make it twice as tall you need to double it a third time, so 32 bricks.

The outside didn’t increase nearly as much though because most of those new bricks are on the inside.

Suppose you need potato peels for a recipe. You want a lot of peel and the interior is irrelevant today. Should you buy one big 8kg spud or eight 1kg ones? It’s the same cost in this example because potatoes are sold by mass not size but it still makes a difference which you choose:

The one big potato will get you 8x the mass of the smaller ones but for your recipe you don’t get 8x as much peel. It’s more like only 4x as much peel. In fact the bigger the potato you buy the less peel you get for your money.

You decide instead to buy a ton of 200g salad potatoes and get a spud ton more peel that way, wasting much less interior material.

When you make things bigger / smaller, their 1D, 2D and 3D characteristics get bigger / smaller at different rates.

This has big practical consequences. Often an exact bigger or smaller version of a thing won’t work exactly the same (or at all).

Shrinking or enlarging humans or animals is a popular theme in fiction — Gulliver’s Travels, Magic School Bus, Honey I Shrunk the Kids, and so on. In reality, the grown or shrunken subjects’ bodies would badly malfunction, and they’d [quickly die in gruesome ways](https://www.youtube.com/watch?v=MUWUHf-rzks).

Let’s say you want to fill a small box with M&M’s as a present for your friend.

First, you try a 1″ x 1″ x 1″ red box.

– Your scoop is just the right size, the box takes precisely 1 scoop of M&M’s.

– The back of the wrapping paper is marked with 1″ x 1″ squares

– To wrap it, you need 6 squares cut out in a T shape [like this](https://en.wikipedia.org/wiki/Cube#/media/File:Hexahedron_flat_color.svg)

Then you decide the 1″ x 1″ x 1″ red box is small. You try a green box instead, its measurements are 2″ x 2″ x 2″.

How much bigger is the green box? It depends on whether you measure its 1D, 2D or 3D characteristics [1]:

– (1D) The green box’s sides are 2x as long as the red box.

– (2D) The green box uses 4x as much wrapping paper as the red box.

– (3D) The green box uses 8x as many scoops as the red box.

[1] Here’s some explanation to help you see why this is so:

– (2D) The green box’s wrapping paper use the same T shape, but each side is a 2″ x 2″ square, which is made up of four 1″ x 1″ squares. Since the 6 sides each have 4 squares, the total squares are 6 x 4 = 24, or 4x the red box’s 6 squares.

– (3D) Think about filling the green box with red boxes. You can put 4 in the bottom layer, and 4 more in the top layer.

Let’s first explore an example, and then I’ll explain the Square Cube law based on what the example taught you

I am in the business of making cubes for a living. I have three subcontractors who work for me:

* Lino, who puts covers on the edges of the cube (1 dimensional),

* Surfacio, who paints the surface of the cube (2 dimensional, the “square” in the square cube law)

* Volumio, who fills the cube with water (3 dimensional, the “cube” in the square cube law)

I used to commission cubes of a consistent size (2 meter edge size). I pay each of these contractors $100 for their work. For each cube, my contractors would need to do this much work:

* Lino has to put 24m of coverings on (because a cube has 12 edges, each 2m) Effectively, Lino makes $4.61 per meter of covering that he puts on.

* Surfacio has to put on 24m² of paint (because a cube has 6 faces, each 2m by 2m = 4m²) Effectively, Surfacio makes $4.61 per m² of paint that he puts on.

* Volumio has to put 8m³ of water in (because this cube’s volume is 2m by 2m by 2m = 8m³) Effectively, Surfacio makes $12.5 per m³ of water that he puts in.

Everyone is happy. But now I want to ship 4m cubes. That’s twice as big. I’m a generous guy, so I’ll offer my contractors $250 per big cube (instead of the $200 that you’d expect because the cube is two times as large).

But still two of them reject this offer. Why’s that? Well, let’s consider what I’m asking them to do:

* For a big cube, Lino has to put 48m of coverings on (because a big cube has 12 edges, each 4m) Effectively, Lino makes $5.21 per meter of covering that he puts on, which is better than the $4.61 per meter he earns for small cubes.

* For a big cube, Surfacio has to put on 96m² of paint (because a big cube has 6 faces, each 4m by 4m = 16m²) Effectively, Surfacio makes $2.60 per m² of paint that he puts on, which is decidedly worse than the $4.61 per m² that he earns for small cubes.

* For a big cube, Volumio has to put 64m³ of water in (because a big cube’s volume is 4m by 4m by 4m = 64m³) Effectively, Surfacio makes $3.9 per m³ of water that he puts in, which is decidedly worse than the $12.5 per m³ that he earns for small cubes.

What happening here is that even though the cube is twice as big, the workloads don’t just become twice as big. Relative to the scaling factor of the cube:

* Lino, being onedimensional, sees his workload grow by the scaling factor (2). He has to double how many coverings he puts on.

* Surfacio, being twodimensional, sees his workload grow by **the square of the scaling factor** (2² = 4). He has to put on four times as much paint.

* Volumio, being threedimensional, sees his workload grow by **the cube of the scaling factor** (2³ = 8). He has to put 8 times as much water into the bigger cube.

As you can see here, depending on how many dimensions you operate in, your workload grows by a different amount for a given scaling factor.

To generalize: the square cube law tells you that when you increase your base number b by some scaling factor, then its cube (b³) will increase much more than its square (b²) will. In other words, the higher the power, the bigger the impact of your scaling factor is.

(continued in comment reply below)

Surface area goes up by a power of two, volume goes up by a power of three. Think about a cube with side length l. The surface area is 6l^2 and the volume is l^3. So for a unit cube surface area is 6 and volume is 1. At l=2 we get 24 and 8. L=3 is 54 and 27. L=4 is 96 and 64. L= 10 gives you 600 and 1000. As you can see, volume increases much quicker than surface area. This has very serious implications in engineering, because certain phenomena, like heat transfer, are dictated by surface area and are acting on volume. So if the system gets too large, cooling it down in any sort of meaningful amount of time can become deeply non-trivial.

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