# Seriously, WTF is up with surface area and volume limiting how big things can grow??

10 views

Disclaimer: I did see a previous question touching on something like this but what I’m confused about was NOT addressed so hopefully this is allowed.

They say that the surface area volume ratio limits how big things can grow because surface area scales as a square while volume scales as a cube, so the ratio of volume to surface area goes up as you get bigger. Fair enough. BUT: how is this not just a matter of what units you’re using?

For example, a 1x1x1 ft cube has a surface area to volume ratio of 6sq. Ft to 1 cubic foot, so 6:1. A 1x1x1 meter cube has a ratio of 6:1 too but the units are meters. Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?

To bring it back to the actual question, wouldn’t your ratio be essentially the same no matter how big your object is? Imagine you expanded everything in the universe by the same amount but kept your unit of measurement the same, you wouldn’t suddenly hit some limit where it stops working right? Does it have something to do with the size of molecules and proteins etc? Please help I am so confused

In: 0 No, because you’ll be using the same units for both the regular size, the square, and the cube. If you use different units for each in order to get the same numbers out it doesn’t change the fundamental fact that heat production in a mammal scales with the cube while its ability to dissipate heat scales with the square, so if you double the size of the creature without making any other changes, it will tend to overheat due to producing 8x as much heat but only having 4x the surface area to get rid of it.

Creatures that don’t produce their own heat, such as lizards, don’t have the same problem, which is why dinosaurs were able to get so large. Creatures that live in the sea are also less affected because water is a much more efficient conductor of heat than air is, which is why the largest animal that has ever lived is today’s blue whale. The surface area to volume ratio is unit-independent, yes. But that doesn’t change the physical meaning of the ratio.

Say that for each cubic foot of living matter, you need 1 pound/min of oxygen. Each square foot can provide 0.3 pounds/min through it. For your 1ft cubic organism, you can provide 6×0.3=1.8 pound/min of oxygen, which is more than the 1 pound/min needed. Good.

Let’s go to the bigger cube. 1 meter = 3.3 ft. 1 cubic meter is 3.3^3 ~ 36 cubic ft, so 36 pound/min of oxygen needed. The surface can provide 6x(3.3×3.3)x0.3 = 19.6 pound/min of oxygen. Not good.

Just playing with units doesn’t change the inherent fact that volume is what determines cellular needs/uptake, while surface area determines nutrient/gas/excretion availability/rate. Since volume scales up faster than surface area, so too does cellular demand scale up faster than nutrient/gas availability. You seem to have a few misconceptions.

Make cubes with dice.

It has a volume of one and width height and length of one with a surface area of six. (6:1)

Make a 2x2x2 dice and it is two wide, two tall and two deep, it has a volume of eight and surface area of 24 (3:1)

3x3x3 is 3 w/d/t and volume of 27 and a surface area of 54. (2:1)

4x4x4 is 4 w/d/t, volume 64, surface 96 (1.5:1)

Kurzgesagt have a great set of videos on the size of life. I think number 2 covers this pretty well. Yes, you could, but the physics aren’t unit dependent. The failures here come from things like “the rate at which one inhales has to grow too high for the tissues of the windpipe to survive” and “bones cannot physically support the larger weight”, and in both cases you’re talking about materials strength. Most of the structural properties of materials have units of force per area (=pressure), and area embeds your length unit into the failure point.

As an example, suppose you have a 1 meter square beam that can support 1 GN/m^(2) (i.e., 1 GPa, but I’m intentionally going to use the explicit force and area units here) of weight, with a 1 meter cube on top of it pressing down on it with that maximum of 1 GN of force. Now, double the size of everything. The cube is now 2x2x2 = 8 m^3 and has 8 times the weight, spread across 4 times the area since we scaled up the beam. That means the pressure on the beam is now 2 GN/m^(2), which is beyond its strength and the beam breaks or buckles.

Now, redefine new units. Introduce the splark (abbreviation sp), where 1 splark = 2 meters. The strength of our beam is now 4 GN/sp^(2). In the initial state, the cube is 1/2 spark x 1/2 splark x 1/2 splark and the area of the top of the beam is 1/4 splark^(2), and the cube still applies 1 GN of force to that 1/4 splark^(2), so it’s applying 4 GN/splark^2 – as before, it’s exactly at the limit of our streel beam. Double the length and we now have an 8 GN force from the cube now pressing down on a 1 splark by 1 splark beam, resulting in an 8 GN/splark^2 pressure and failure of the beam.

In either case, the beam fails above a volume-to-area ratio greater than (1 m^3 / 1 m^(2)) = 1 meter (the initial state), which is equivalent to ((1/8) sp^3 / (1/4) sp^(2)) = 1/2 splark. The value of the ratio embeds your unit, and the system fails at the same value (expressed in your unit of choice) regardless of measurement choice. You’re not using units quite right. You have to compare apples to apples. Let’s use two different units to prove this:

You have a one meter cube. You double its size, and have a two meter cube. Its surface area goes from 6 square meters to 24. Its volume goes from 1 cubic meter to 8. Its volume has increased twice as much as its surface area.

Same cube, let’s use feet instead. 3.28 feet is one meter. The 1M cube has a side length of 3.28 feet. This makes for a surface area of 64.55 square feet, and a volume of 35.287 cubic feet. Scale it up, just like last time, to a side length of 6.56 feet. SA is now 258.2 square feet. Volume is 282.3 cubic feet. If we compare the two, we see that the volume has increased twice as much as the surface area.

This holds true for any unit you choose, so long as you don’t goof up the math during your conversions. The most common such error is forgetting to square during unit conversions. Example:

1 square foot = 12 square inches. This is wrong. A foot is 12 inches, so a square foot is (twelve squared) square inches, or 144 square inches. Yes, you could. However, because they’re in different units, you would need to convert between them before you can get any meaningful information out of it. If you invent a new unit for each different edge length of a cube, then you have an infinite number of different units, and if you wanted to say anything like “cube X is twice as big as cube Y” you’d first need to re-measure cube Y using the same unit you measured cube X in.

So, a 1x1x1 ft cube does have a total surface area of 6 square feet, and a 1x1x1 meter cube has a total surface area of 6 square meters. But a 1x1x1 ft cube is also a 0.3×0.3×0.3 meter cube with a total surface area of approximately 0.55 square meters, demonstrating that this 6:1 ratio is really just an illusion of the chosen measuring system, not an actual fact of reality. Remember, units are just something we use to be able to talk and think about reality in convenient ways. Measurements aren’t real. The only real thing here is the fact that the surface area of a cube increases exponentially as you increase the edge length linearly, and this is true in every unit you use to describe a cube increasing in size. Surface area is a large limiting factor for life. And note that surface area is measured in square units, volume in cubed. So when you convert from one unit to another, you need to take that into account. Changing the “starting scale” also changes the ratios.

Say a meter is equal to 3 ft(approximate so that we get whole numbers). A square meter is 9x a square foot, for a total of 54 square foot. A cubic meter is roughly 27x a cubic foot. That’s a area-to-volume ratio of about 2:1, down from the 6:1 ratio of the foot measurements.

Additionally, some processes in bodies don’t scale up or down infinitely. Capillary action stops working in tubes above a certain size. Vacuums cannot lift water above about 14m. This is all independent of units.

To simplify the whole thing imagine a weight suspended from a rope. The rope is just strong enough to not snap under the weight.

How much weight a rope can carry is largely depended on its diameter.

If you get a rope that is twice as thick in diameter it might carry about four times the weight.

However if you double the size of a weight it will weigh 8 times as much.

For living things that means that if you for example want to scale a human body up to twice its height the bones would be twice as long and have cross section four times as large but their weight would be eight times as much.

A giant human would need much thicker legs than a simple scaled up normal size human.

You also run into problem in other areas like all that stuff about lungs and the heart and the blood and heat dissipation.

There is a reason why large animals look different than small animals would if you scaled them up. The problem is that the ratio changes as the volume grows, even if you stick with the same units. A 1x1x1 meter cube has a 6:1 ratio, yes, but if we give the cube twice the volume (so it’s a 1.26×1.26×1.26 meter cube now, roughly), then its surface area ends up around 9.5, so the ratio is now something like 4.76:1

Sure, you can say “yes, but if I make up another unit then the ratio between *those* units is what I want it to be”, but that doesn’t change the fact that as the object grew, the ratio between volume and surface changed.

Physics doesn’t care which unit you choose to use. It cares that “compared to when the object was smaller, it now has a smaller surface to volume ratio”

The problems this causes are varied, but it’s not as simple as “suddenly the molecules in my body magically stopped working”. A few examples I can think of are:

1. because elephants are so big, they have volume many, may times greater than that of a human, but their surface area isn’t *as much* greater. This means they already have trouble cooling off. They generate heat throughout their massive bodies, but the heat can only escape through their skin, and compared to how big they are, they don’t have *that* much skin. So if you made an elephant with twice the volume, it would produce twice the heat, but it would not have twice the surface area from which that heat could escape. It would die.

Airplanes would also have problems if you just made them twice as big without changing anything else. Because their volume (and their weight) would double, but the surface area of the wings would not. The wings would need to be made *even bigger*. And of course, even bigger wings would suffer greater stresses and would need to be made stronger.

Plants can only absorb sunlight through their surface area. So if the volume of a plant doubled, but its surface area didn’t, it’d suddenly get relatively less sunlight than it did before. > For example, a 1x1x1 ft cube has a surface area to volume ratio of 6sq. Ft to 1 cubic foot, so 6:1. A 1x1x1 meter cube has a ratio of 6:1 too but the units are meters. Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?

This number is not *unitless*. So it’s arguably not accurate to even call it a *ratio*. Your process of reasoning only works if you can cancel out the units on both sides. But when you have “6 square meters : 1 cubic meters”, the best you can do is cancel it to “6 : 1 meter”, or “6 *per meter*”. (Indicative, in this case, of how much extra area you’re getting per specific depth of tissue)

Somehow along your argument you seem to have omitted this unit without justification, which is the reason for your confusion. The thing you’re forgetting is how area and volume related to the ability of some structure to hold its own weight.

Volume correlates with weight. Switching units of volume isn’t going to change its weight. You can’t make an object lighter or heavier by deciding to measure its volume in different units.

The internal strength of an object correlates with area (specifically cross-sectional area). Switching units of area isn’t going to change its strength. You can’t make an object stronger or weaker by deciding to measure its area in different units.

As an object increases proportionately, its volume (and therefore its weight) increases by the cube of that factor but its area (and therefore its strength) increases only by the square of that factor. So its weight increases much faster than its strength, meaning at some point its weight will exceed its strength. > Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?

Yes, but the laws of physics don’t depend on the units you choose to work with. Just because 1km is a smaller number than 10ft doesn’t mean it’s a shorter distance. The actual numbers are just arbitrary. This is why physicists often prefer to work with [dimensionless quantities](https://en.wikipedia.org/wiki/Dimensionless_quantities) which are independent of the units used to measure them.

Any particular material has something called it’s yield strength. That’s the maximum amount of stress (i.e force per unit area) that it can take before it is permanently deformed. If you double the size of an object, the volume becomes eight times larger (2^3), so the object now weights eight times as much. However, the cross sectional area of the object is only four times larger, so the material can only take four times as much load.

That means that as you scale an object up, the weight it has to support grows faster than the ability of the material to support that weight, and eventually you reach a point where the material cannot support the weight and will fail under the load. > For example, a 1x1x1 ft cube has a surface area to volume ratio of 6sq. Ft to 1 cubic foot, so 6:1. A 1x1x1 meter cube has a ratio of 6:1 too but the units are meters. Couldn’t you always define your units so that you have a 6:1 ratio with any size of cube?

Sure. But we’re concerned with *growth*. You can use whatever units you want, but a 2x2x2 <unit> object is always going to have 4x the surface area of a 1x1x1 <unit> cube, and 8x the volume.