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I know about the square-cube law, and this question is not whether dragons can fly at all, but how high they can fly. This link may help deal with some of the problems that come up.

Edit: Assume their bones are strong enough for whatever is necessary (thanks, Tim Hansen!) and the dragons are able to flap their wings but spend more time gliding. If for your answer to work the dragons' bodies would need to be smaller, feel free to take liberties.

In movies such as How to Train Your Dragon, Eragon, and Pete's Dragon, dragons are able to fly above (at least some) clouds. Knowing that there is less air resistance and less air to breathe, and knowing that the few birds that do fly above the clouds do so for limited time, this seems unrealistic.

Assume that the dragons' bodies are about the size of a horse, the wings are as big as needed, and the dragons take off by leaping off a cliff.

If some sort of lifting gas would help your answer, that's fine too.

How high can dragons fly (and how can they breathe at that height)?

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closed as primarily opinion-based by Morris The Cat, A Lambent Eye, Mołot, Cyn says make Monica whole, We are Monica. Sep 26 at 15:14

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ If we don’t know how dragons fly, there’s no way we can tell how high they can fly. You’ll have to give us the details of your fix for the square-cube law to get any sensible answers. $\endgroup$ – Mike Scott Sep 25 at 5:26
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    $\begingroup$ Lifting gas never helps, unless it's either actually anti-gravity gas (in which case it really should be liquid) or your dragons are big fat blimps. $\endgroup$ – ikrase Sep 25 at 7:21
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    $\begingroup$ @ikrase - Note that the 1979 book "The Flight of Dragons" posits precisely that - that dragons are big, fat, hydrogen-filled blimps - and then derives most of the mythological characteristics of western dragons (fire breathing, corrosive blood, hoarding gold, etc.) from that premise. $\endgroup$ – Dave Sherohman Sep 25 at 7:48
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    $\begingroup$ Clouds can be quite close to surface (or even can lay on surface). Typical low layer cloud cover is at about 200m - 1 km. And is achiveable by any self-propelling flying creature (even by veloplanes) $\endgroup$ – ksbes Sep 25 at 12:38
  • $\begingroup$ This seems primarily opinion based to me. The answer depends entirely on the answerer coming up with their own principles of how dragons ignore the laws of physics. I'm voting to close as POB pending clarification from OP. $\endgroup$ – Morris The Cat Sep 26 at 13:45
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Let's first look at how high a bird can fly:

The endangered Ruppell's griffon vulture is the highest flying bird ever recorded, and it can ascend to heights of 37,000 feet. That's the average height at which a commercial airplane travels (and a Ruppell's griffon vulture has indeed been sucked into a jet engine at 37,000 feet)

Second comes the common crane, with 33000 feet.

I guess that large wings and the sapient use of soaring winds greatly assist in reaching those heights, a stunt that also your dragon might be able to perform. Same goes with breathing, birds have a more efficient respiratory system, so the same can have your dragon.

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  • $\begingroup$ The common crane does it briefly to cross mountains, but they don't maintain that altitude. $\endgroup$ – Separatrix Sep 25 at 7:01
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Using explanations like gas filled cavities and such, like Naomi Novik does, is not the best approach. Such a dragon would not look like a dragon, but a hot air balloon. If the explanation makes it even less credible, don't try to explain it at all.

(Then there is Falcor from Neverending Story, who doesn't have wings at all.)

One of the best ways around the square cube law is perhaps this suggestion:

With strong enough bones, dragons really could fly. 1

For dragons to fly they would need super strong magic bones. For something to move in one direction there must be a force in the other direction. When you swim you push the water back with the same amount of force that you go forward. For something to fly it must push air down with at least as much force as it weighs, i.e. when a plane flies the shape of the wing pushes air down with enough force to lift the plane. In the case of winged dinosaurs, they would use their huge muscles to push the air down with the amazing amount of force that would been needed to lift their huge frame; Pterosaurs did this very effectively and could get quite large. The largest species, Quetzalcoatlus northropi, was around 550 lbs.

The limiting factor in pterosaur flight wasn’t their muscles, it was their bones. Any bigger than Quetzalcoatlus northropi and the force needed to push down is greater than the force needed to break a bone. Something the size of a dragon would require generating so much force to lift that the bones in his wings would break. If, however, the magic of dragons was housed in their bones, they would be able to fly.

1 Rebecca Thompson - The Science of ‘Game of Thrones’: 10 Surprising Facts Behind the Fantasy

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Something like a swim bladder would help but fill it with helium, hydrogen or another gas (you would away to expel it for landing and pull it back in). If this is fantasy you could slap magic on it, think Chinese dragon they are giant serpents but they still fly. If you don't want to use magic then here are some numbers that might help you. Birds can go up 5,000 feet using winds to help. Helicopters can go up 10,400 with a really good one going upwards of 13,800. Your wing size is important for this because the equation to calculate life is Coeeffecient X Density X Velocity squared X Wing area, all over two L = Cl * A * .5 * r * V^2 these are all factors of how high you can go also Human cannot go above 26,000 feet at all without acclimation though we pass out at 15,000 if you give more information about density of the dragons and the wing size which does matter. I'm sure we could help more

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  • $\begingroup$ Hydrogen would be better as it's easier to come by as a natural byproduct. However it wouldn't need to be expelled, you could get by with compressing it. $\endgroup$ – Separatrix Sep 25 at 7:03

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