Dragons.

They might possibly be the most popular cross-cultural mythological creature in the world. While I won’t pretend to have a real count on ancient mythologies proving this, most of us would agree that dragons are easily the most common mythological creature portrayed in modern media.

Now, let’s be clear, I’m not saying dragons ever existed. Others have done research on the plausibility of real dragons in history, but come on. I’m a fantasy novelist. I don’t care if they really existed, I just want to know if they are possible.

I’m talking your typical, gigantic, plane-sized dragon here, not a tiny flying reptile that is a dragon in name only.

So, could a dragon actually fly?

The Problem

First, let’s establish the problem. Why can’t animals (be they birds, dinosaurs, dragons, or fairies) grow bigger and bigger, as long as they scale proportionally?

In its simplest form, it’s for the same reason a snowman takes longer to melt than a tiny snowball: the area to volume ratio. This is called the Square-Cube Law (hereon abbreviated as SCL).

The end result is that wing length must increase at a rate 1.5 times higher than the length of the animal. If that’s enough for you, feel free to scroll to the next heading. Otherwise, here’s the explanation:

For our calculations, let’s define “bigger” in terms of the mass (m) of an animal.

Assuming a constant density (i.e., mass is directly proportional to volume), an animal that grows completely proportionally in all directions, will not have a constant mass to area ratio. Rather, the ratio of area to volume will decrease at a rate of 1/x^(1/3), where x is the mass (in other words, if l represents the characteristic length, area grows at a rate proportional to l², and volume l³, so the surface area to volume ratio decreases at a rate of 1/l).

This matters because, as we’ll see later, the area of a wing is directly proportional to how much weight a flying animal or machine can lift off the ground.

When we graph this, we see that wing length must increase at a rate 1.5 times higher than the length of the animal (assuming the rest of the body retains a constant shape). This works fine for swallows and eagles, but when you get to Smaug’s size? That’s questionable.

 

 

Existing Possible Solutions

Given the natural limitations imposed by the SCL, most authors recognize that they must make the underlying assumptions of this law (a constant density and shape) inaccurate. They devise methods of making them lighter, so they can maintain their desired aspect ratios without having a mile-long wing that would require ridiculous torque not to snap in half.

Here is a (not at all exhaustive) list of the solutions out there:

• Game of Thrones by George R. R. Martin – no attempt at rationalization
• The Hobbit by J. R. R. Tolkien – no attempt at rationalization
• Eragon by Christopher Paolini – no attempt at rationalization
• His Majesty’s Dragon by Naomi Novik – lighter than air sacs
• Dragon Keeper by Robin Hobb – no attempt at rationalization
• A Natural History of Dragons by Marie Brennan – exceptionally light bones, plus special feathers
• Dragon Planet by Dan Wells – different planet, different physics
• Discworld by Terry Pratchett – rocket-propelling fire coming out of its hindquarters
• Reborn: As a Defective Drake by Zoe Townley – multi-dimensional magic dragons
• Aithos by Izaic Yorks – light bones, internal air sacs, and some transubstantiation / magic (publication date: November 2024)

Now, I’m not saying that authors have to or even should rationalize away their dragon’s magic, and as you can see by this list, many of the more famous dragon books do not attempt to do so. But for those that do, let’s see how their explanations hold up.

Methods for Decreasing Weight

Birds: The Real Dragons

Since flying living creatures do actually exist in our world, it makes sense to use them as a place to start. Birds support hollow bones and air sacs (among other adaptations) to minimize their weight.

They are on average 0.602-0.918 g/cm3, which means they are roughly 9%-40% filled with air (we’ll get into that math later).

Even with these adaptations, flying birds have a size limit. The heaviest flying bird (that isn’t extinct) is the kori bustard, which can get up to 19 kg.  

So, it’s natural for authors to see how they can exploit and extend what birds have already figured out to make their dragons fly.

Before we move on, I’ll briefly address the dinosaur in the room.

Yes, there were larger winged dinosaurs. The Quetzalcoatlus had a wingspan of up to 12 m (36 ft). Its weight can only be estimated, and these estimates have ranged from under 200 lbs to up to 500 lbs. But, scientists aren’t sure how–or if–the Quetzalcoatlus could actually fly. Current theories suggest they only flew in short bursts and/or made use of thermal or dynamic soaring. So, they may have flown…or maybe not quite. Even if we pretend that a Quetzalcoatlus could out distance a common swift, though, it’s still much smaller than most proposed dragons. 

I considered going into more detail about dinosaurs throughout this post, but it was getting too long, so I decided to draw the line. 

Lighter Bones

This I’ve only seen twice, but since the math is simpler, we’ll start here. A Natural History of Dragons and Aithos feature dragons with exceptionally light bones. Whether bones such as they are described in the book are biologically possible is out of scope for this article, so let’s assume they exist.

Here’s the problem. Bird bones only make up approximately 6%-8% of the total mass of the bird. So, if we half the density of the bone (which feels generous to me), we’ll say you can decrease the mass of the dragon by 4%. Thus, since characteristic length is proportional to the cube root of mass, the length of the dragon can increase by roughly 2%.

If we go out on a limb and say these magic bones have negligible weight (8% lighter), your dragon can be 3% longer. Of course, longer wings leads to larger wing area, which leads to more Lift (more on that later), so this doesn’t hold exactly. Still, the gain is limited, because longer wings have increased strain, so you can’t go on lengthening the wings indefinitely.

If I’m being supremely generous, I’d say this method gives an animal no more than a 15% increase in length, reasonably. That’s not nothing, but it’s not that much.

Both of these authors did hint at several other adaptations the dragons have to maximize lift, which I’ll mention later, but even so, lighter bones doesn’t break the glass ceiling on weight, it only stretches it slightly.

Air sacs, and Lighter-than-air sacs

A common solution I see proposed for this is the air-sac, or even the lighter-than-air-sac. Interestingly, while this is a solution I commonly see proposed on forums and discussion boards, I’ve only seen it twice in a book: His Majesty’s Dragon (and later books in the series) and Aithos.

Again, this is an extension of the anatomy of birds.

First things first: Does a lighter-than-air-sac make sense?

Spoiler alert: no. Unless by “dragon,” you mean “blimp.”

What, exactly, do the air sacs do, and what would be the impact of replacing that air with something, well, lighter?

I understand the impulse. It would make sense that if something had a bunch of helium in it, it’d float easier, just like a balloon. But to really understand what’s going on here, we have to look at the forces that impact an object in flight.

For a dragon to fly, the upwards forces must at least equal the downwards forces. In this example, I’ve outlined the main forces acting on a dragon (or any body) in flight. The forwards-backwards motion doesn’t matter, here, so we will ignore them for now.

The only downward force is gravity. We’ll start there. Gravitational force is equal to mass times the gravitational constant, or density times volume times the gravitational constant.

So, if we calculate the average density, we can see how the force of gravity relates to volume.

Like I mentioned, birds have air pockets in their body, and as such, they are much less dense than the average animal. For simplicity sake, we will divide the flying animal into two parts: the “flesh” or body part, and the air part.

The average density of the animal can be calculated with the following equation:

We’ll approximate the density of the ‘flesh’ portion as the density of water (62.4 lbm/ft3). This is reasonable since most of the body is made up of water.

The density of air, comparatively, at standard temperature and pressure is 0.0763 lbm/ft3.

When you plug in those numbers, you see that the density of the air portion is negligible, basically zero, so we can cancel that out.

Hold up, then! You just said the air is approximately zero?! But what if I make my gasses lighter than air? Doesn’t that change anything??

No. Sorry. Negligible is negligible. Unless you have like a lot a lot of air (or the replacement gas), and in that case, I would call your dragon a blimp instead.

Okay, I hear you. But what if we make the air sacs BIGGER?

Sure, let’s finish our density calculation.

When we simplify everything, we see that the density is directly proportional to the ratio of Vflesh to Vtotal.

And, plugging that back into the gravitational force equation, the gravitational force can be simplified to 62.4 lbf/ft3*Vflesh.

Since we’ve established air sacs don’t influence the downward force, we’re left with the upward forces. There are two: lift and buoyancy. When someone talks about an air sac and they’re imagining the dragon floating along, what they’re really talking about is increasing buoyancy (it has no impact on lift; we’ll get there later).

Buoyant force is calculated by Archimedes’ principle, shown below.

Does it look familiar? Here, Fb is the buoyant force (the negative means in the upwards direction), ρf is the fluid density, g is the gravitational constant, and V is the total volume of the fluid that is displaced (i.e., the total volume of the animal).

Like I mentioned earlier, birds are on average 0.602-0.918 g/cm3, and using the density equations we just derived, we find that they are typically 9%-40% filled with air.

Let’s look at the most extreme possible case of lighter-than-air-sacs, and say that the space inside the sacs has a density of 0 (a complete vacuum).

Let’s give our dragon slightly larger air sacs, and call him 50% air, and see what we get.

So, if a dragon is half air, the buoyant force only accounts for 0.2% of the total force required.

Let’s make the air sacs…bigger. Where the dragon is 10x the size it really needs to be.

Now the buoyant force is 1% of the total force required.

So…basically, air sacs don’t help. Air sacs only help when they are replacing unnecessary weight inside the animal to minimize the gravitational force.

Magic Dragons and Other Explanations

Adding a New Upward Force

This, rather silly, explanation is the route Terry Pratchett took in designing his Discworld dragons. Disclaimer: I haven’t actually read his books with dragons, so this is hearsay. His dragons have rocket-propelling fire coming out of their hindquarters.

This is actually not a bad option. It is, of course, the way rockets and planes fly. I’m not going to waste time doing math on it, because it would work, if you make the rocket strong enough. How such a system would be biologically possible is another story, and out of scope.

Multi-Dimensional Magic Dragons

In Reborn: As a Defective Drake, dragons (and other very large creatures) are able to tap into additional dimensions beyond those the mere humans can touch, including one with very low gravity. By shoving a portion of their mass into that low-gravity space, dragons no longer weigh as much in this dimension. Thus, they can achieve flight with lower lift required.

Now, I am not a theoretical physicist, so I may be missing some things in my analysis, but I find this to be a rather clever solution to the problem. Still, since science has not discovered any way to tap into extra dimensions (yet), this boils down to “magic dragons.”

My immediate concern with this explanation is how the mass within the low-gravity dimension is lifted. Does the dragon need to provide enough lift in that dimension to offset its mass? If so, is there enough air resistance in a low-G atmosphere to provide that lift? Or is the dragon somehow able to redirect the upward force in this dimension to the extra dimension to achieve flight? Since we’ve crossed into the realm of made-up-physics, the answers are almost irrelevant.

Still, magic, multi-dimensional dragons sound pretty cool to me!

 

What about dragons not on earth?

A case study: Dragon Planet by Dan Wells

This book takes a different approach to modifying the forces. It takes place on a different planet with a different atmosphere (and presumably gravitational forces, but I don’t remember the specifics of those).

Let’s run the numbers for these.

Sadly, I don’t remember if the book described the exact composition of the atmosphere, and I don’t have easy access to it, so I’ll be making a few more assumptions than I otherwise would be.

Xenon makes up some of the atmosphere of this planet. If we make the random assumption that xenon represents the same proportion of the air that nitrogen does here (and the rest is, more or less, oxygen), the gas behaves ideally, the gasses don’t settle into separate layers based on mass, and the gravity of the planet is approximately the same (guys, this math got real shady real fast), we can say the buoyant force on the Dragon Planet (yeah, I don’t remember the name of the planet either) is approximately 3.8 times higher than here on earth.

That is still not very much.

But, hold on. We still have another force to contend with. Let’s take a look at Lift.

For constant air conditions and body shape, the equation for the lift force is as follows:

Cl is the lift coefficient, which has complex dependencies on air viscosity and compressibility, and the shape of the body.

ρf should look familiar by now: the fluid density. v is now velocity (not volume anymore), and A is the area of the wing.

Let’s assume for now that these dragons are otherwise the same as our dragons (A and v are the same; and any dependencies of Cl on the shape of the body are the same).

Outside of the dragon, ρf matters a lot more. In the lift equation, the lift force is directly proportional to fluid density. Therefore, on this planet where the atmosphere is 3.8 times denser than here on earth. I’m not quite convinced that will translate linearly into 3.8X smaller wings (by area) being sufficient for flight, but you could certainly get by on smaller wings, all else being equal.

That’s a big difference!

Maximizing Lift Here on Earth

Assuming you want to stay on earth, let’s look at the lift equation again to see how we can make our dragon fly.

Area

The first is a larger area. As long as the wing joints and shoulders can support the wings, larger wings is the way to go. The problem is, wing joints and shoulders can’t provide infinite torque. I think it’s safe to say the birds we have here on earth have approximately the highest wing area to body length ratio as is biologically reasonable.

Shape

Another, less obvious way, is optimizing the body shape, which impacts Cl. I’ll be honest, I’m not sure how optimized birds are for flight. Maybe they are extremely optimized and there’s really no way to make a living creature fly better. Or maybe a few tweaks could make flying easier for them.

This is what A Natural History of Dragons attempts to do with this mention of feather shape: “The roughness on the underside of the wing comes from tiny scales, which are not present on the upper surface. These cover tiny holes that perforate the wing, and are hinged to form a sort of valve. When the wing lifts, the valves open, reducing the resistance the dragon’s muscles must overcome. When it sweeps down again, the valves close, allowing the stroke to have its fullest effect.”

I will be honest, I’m not entirely sure what this paragraph means or how to picture what is happening here. I’m not an expert on aerodynamics. But, I’d venture to say this is again just stretching the glass ceiling a bit. Maybe you’d get 10% extra lift, maybe 20%. But probably not 100% more.

Speed

But the final, and my personal favorite, way to make the dragon fly is through velocity. Planes are, I’d say, heavier than dragons, and yet, they fly with incredibly low wingspans. How? Velocity. They get on a runway and go, go, go until the lift overcomes the weight and the plane lifts off the ground. According to the ever-so-reliable Wikipedia, take off usually occurs at 150 – 180 mph.

I think our dragons can do a bit better than that, with larger wings and lower weights.

Let’s say we give a large cheetah (140 lbs, more than 3x heavier than the heaviest flying bird) a pair of dragon wings and let him run. How fast would he have to go to get off the ground?

Or maybe the dragons could have a downhill ramp takeoff. Dragons are frequently depicted as intelligent creatures. Can they make a platform with wheels? They hop on, ride down the slope, and take off somewhere along the way. Or climb up to a cliff and jump off.

Maybe it’s not exactly the majestic picture we have in mind when it comes to dragons, but, hey, whatever gets them off the ground.

 

Conclusions

There are probably better solutions than that out there, but, I hate to break it to you, lighter-than-air sacs isn’t one of them.

Or, we can just call them magic and not explain it.

While I might snicker about how ridiculous the math is, His Majesty’s Dragon is my all-time favorite dragon book (and series). Not everything must be realistic for it to be enjoyable.

Still, I hope you found this blog post enlightening. If nothing else, it goes to show why we don’t have gigantic creatures flying around in real life—at least here on earth. As Dragon Planet shows us, a denser atmosphere has a strong impact on the ability of a dragon to fly.

With no further ado, the Most Realistic Dragon award goes to…

(Trust me, it’s a super official award)