http://tvtropes.org/pmwiki/pmwiki.php/Main/SquareCubeLaw

With 1/27 the volume but 1/9 the surface area, the smaller cube will be three times as agile as the larger cube. (Assuming they are capable of movement.)

— Joe Walcott "The bigger they are, the harder they fall."

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A scientific principle often ignored in media:

When an object undergoes a proportional increase in size, its new volume is proportional to the cube of the multiplier and its new surface area is proportional to the square of the multiplier.

For example, if you double the size (measured by edge length) of a cube, its surface area is quadrupled, and its volume is increased to eight times its original volume.

The point of this law is that with living beings, strength is (more or less) a function of area (the strength of a muscle or bone is proportional to the area of its cross-section, not to its total volume), but weight is a function of volume. And Newton's famous Second Law (the "force = mass × acceleration" one) means that if you double a critter's height while keeping it the same shape, you end up with four times the muscle power moving eight times the mass, so instead of having the same relative agility as the original, the double-sized creature actually has only half. The same goes for most machinery.

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This applies to flyers as well: Double the size, and you get four times the wingpower attempting to keep eight times the weight airborne, so the creature's ability to fly has actually been cut by half. Helicopters are hit particularly hard by this law; the largest payload of a cargo helicopter is about 20 tons, versus the world's largest airplane, with a payload of 275 tons. Sorry, Hotelicopter.

Airships, on the other hand, benefit greatly from the square/cube law, as even small increases in size can quickly increase the volume of buoyant gas they can carry. Take the Graf Zeppelin and its successornote and The Hindenburg's sister ship the Graf Zeppelin ll for example. The original Graf was 776 feet in length. The Graf ll was a mere 30 feet larger in any direction, but carried double the volume. Because these gains came at almost no increased structural weight, the returns went entirely into making the Graf ll an even more palatial flying cruise liner than her predecessor.

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Buoyancy in general, whether it be in the air or in the water, is an easy way to minimize the limitations of the square-cube law when increasing size, because buoyancy is dependent on density, not mass. Good news for the whales, then.

This law is relevant when an object is shrunk down as well. Make something half its size and it will have roughly twice the proportional strength and endurance. This is why small animals, like ants, are able to carry things far heavier than themselves, part of how fleas can manage to jump so far relative to their size, and why cats can survive falls of effectively any height — get small enough and your terminal velocity will be a survivable speed. However, don't think this is all win for the Incredible Shrinking Man, who will likely not survive to enjoy his new found strength. Since body heat production is at least partially proportionate to volume, while heat loss is dependent on surface area exposed to the air, a shrunk human will find they are dissipating heat faster than their body produces it. It won't be long until the shrunk human freezes to death, even during a summer day. (This, incidentally, is why shrews eat their body weight in food and hummingbirds live on sugary nectar — they need such high-energy diets to avoid this.)

A full explanation for the biological aspect is a lot more complicated due to subtler factors (muscle/bone stress, required oxygen uptake, dissipating body heat, etc.), but the gist of it is the same in every case: You can't just scale something up (or down) to a different size and expect it to still work the same way as it used to.

Again, the law is not limited to living creatures, but applies to anything with mass (and, well, everything has mass): A skyscraper twice as wide and tall as another will have eight times the weight, and require a far stronger support structure — wood and brick just can't hold the weight (traditional wood, that is). This is why modern skycrapers were impossible until steel could be mass-produced to build their frames. Likewise, the humanoid Humongous Mecha needs incredibly strong legs to hold its massive frame upright (probably some sort of Unobtainium), and that's not even considering how the ground beneath it also needs to be able to support that same amount of weight without caving in, or the fact that it needs some incredibly powerful motors just to get those powerful legs and arms moving (which is why we call them Impossibly Graceful Giants).

Knowing that audiences are becoming more savvy about this as compared to the days when Attack of the 50-Foot Whatever and the Incredible Shrinking Man were safe, standard plots, many creators who knowingly break the law will try to invent some Artistic License  Physics to justify or Hand Wave how their creation can get away with breaking it — say, the Applied Phlebotinum didn't just change their size, but also does something else to sustain their new size and counter the Law's negative effects upon them.

See Attack of the 50-Foot Whatever and Humongous Mecha for examples of media ignoring the Square-Cube Law. Sometimes justified by the use of Required Secondary Powers. Compare Muscles Are Meaningless and Pintsized Powerhouse. Compare/Contrast Giant Equals Invincible, where the law can either debunk or justify the trope (as a giant creature super-strong and massively-durable enough to simply withstand its own weight and be able to move with any reasonable efficiency would logically be able to shrug off far more damage than something smaller). Do not confuse with Scaled Up — though the trope name may sound familiar, Scaled Up involves a serpentine transformation that usually completely ignores the laws of physics, anyway.

Examples:

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Pinballs

The backglass for Shaq Attaq shows Shaquille O'Neal as 30 times larger than the other players, effortlessly crossing the court in two strides while holding a minuscule basketball between his thumb and forefinger. Needless to say, the Square/Cube Law doesn't even get a passing glance.

Pro Wrestling

André the Giant was so huge due to acromegaly, a disorder of the pituitary gland which essentially causes your body to try to violate the square-cube law. He spent the later part of his life in constant pain due to the strain on his muscles and circulatory system, and died at 47.

Tabletop Games

GURPS details the stats needed for monsters of various impossible sizes with an eye toward the Square Cube Law (how the creatures can be so big is left up to the GM).

Dungeons & Dragons: It is mentioned that creature who has his all dimensions doubled weighs eight times as much, but avoids the rest via A Wizard Did It. Notably with the spells Reduce Person and Enlarge Person, for which there is an equal increase/decrease in Strength and Dexterity, respectively, but only minor ones considered the shift in mass (mainly for balance reasons). Somewhat ignored in that a creature's lifting capacity does not typically scale directly with the creature's mass. For an 8 times mass increase for an extra size category, a creature typically receives a 6 times increase in carrying capacity. It gets worse when A Wizard Did It, as spells such as Enlarge Person increase the creature's weight by the usual eight times, but only increases their carrying capacity by approximately 2.6 times. There is an artifact in the Book of Vile Darkness called the Despoiler of Flesh that gives you very flexible control over a creature's shape, but more or less points out that it has to be scientifically plausible or the creature will die because of an unsound anatomy. In editions where flight maneuverability is mentioned, it's noted that the largest dragons become far less agile as they grow bigger. In 3.5, for instance, Small or Tiny dragons have Average maneuverability, Medium, Large, and Huge dragons have Poor maneuverability, and Gargantuan or Colossal dragons have Clumsy maneuverability — basically, the largest dragons will bank like cruise liners in flight, and are completely unable to pull off things like hovering or flying backward. And while they do get faster, it's nowhere near proportionate to their size; a dragon the size of a sperm whale travels only twice as fast as one the size of a border collie. Of course, the fact that said dragon is able to get off the ground at all is violating this trope pretty hard. One sourcebook explains that dragons get off the ground through a mix of ridiculously strong wing muscles, surprisingly lightweight bodies, and the fact that dragons are partly elemental (hence the Breath Weapon), which gives them a lot of energy to spare. Older game supplements often gave the existence of massive creatures like giants, rocs, and Big Creepy-Crawlies a handwave by stating that such creatures were connected to the innate magic field of the planet that helped sustain them beyond the limits of what bone and muscle alone could accomplish even if the being had no other magical abilities. It was suggested that were they were to find themselves transported to a world that lacked any innate magic (which was much stronger than what the Anti-Magic effects in the game), they'd instantly perish.

When RoleAids released Giants, a vintage third-party D&D supplement, they took this trope into account, rationalizing giants' physiology with honeycomb-framework bones, radically different leg musculature, and super-tough hide to contain their extremely high blood pressure. Oh, and a heaping dose of A Wizard Did It (or rather The Gods Did It) for titans.

Star Wars d6: The Executor class command ship is roughly 12 times the length of the Imperial Star Destroyer in each dimension, or roughly 1728 times its total mass and volume, and roughly 144 times its total surface area, but only accommodates roughly eight times its total crew complement (including stormtroopers and TIE fighter pilots) and 12 times as much weapons payload.

Warhammer 40,000: The Imperial Titans and Baneblades, and Gargants, and Squiggoths, and Hive Tyrants... thanks to 40k's Rule of Cool-powered physics, it's likely cumbersome size can always be counterbalanced by the number of guns bolted to it. Taken to hilarious extremes with the Hierophant Biotitan who, despite being as large as a warhound, is supported by 4 relatively tiny stalks it calls legs. How it doesn't explode from its own weight is anyone's guess. note It's implied that the Hierophant is a very powerful Psyker, but has to dedicate all of its psychic power to maintaining a Warp Field, giving it a powerful armor save while at the same time maintaining its structural integrity. . Unsurprisingly its model is one of the most fragile non-dark eldar ones in the whole line. Averted in a few places, though. Space Marines are about 8 feet tall and almost 8 feet WIDE, with numerous extra organs, and a giant suit of power armor to help take the strain of their larger proportions. Of all the factions to get this right, the ORKS versions of Titans (called Gargants) seem reasonable. Either they have very widely spread out feet, or tank treads to move about, and it appears most of the weight is concentrated towards the ground. Also, if anyone has the right to ignore the square cube law, it would be the Orks, since their technology relies heavily on their innate Psychic Powers to work in the first place. The Square Cube Law is actually referenced by Imperial Biologists, who are astounded that creatures the size of a Squiggoth can exist (in their own words, it should collapse under its own weight and be completely incapable of movement, though such knowledge is likely cold comfort to troops staring down one charging at their lines). The only other faction to field super-huge creatures is the Tyranids, though both groups have in-universe explanations (the Tyranids make use of genetic engineering and Bizarre Alien Biology; the Orks just get to fall back on their usual excuse of "thinking it's real makes it real", which explains why Squiggoths only show up in large Ork warbands). The Tau initially only fielded relatively tiny Mini-mechas in the form of Battlesuits; one-pilot mechas roughly 10-12 feet tall to carry heavy weapons on an infantry scale. They scoffed at the idea of Imperial Titans, whom they thought that the huge amount of resources and manpower to keep one of those things running proportional to just, say, bring a massively huge gunship or artillery is just ludicrous and chalked it up to Imperial Propaganda. One Damocles Gulf War later they realized the humans were crazy enough to make titans and despite the impracticality of it, their effectiveness were unparalleled. This led to the development of the Riptide, Stormsurge, and titan-esque Ta'unar Supremecy Armors

Werewolf: The Apocalypse fails to take this trope into account with the Garou and other shapeshifting races. A Garou's crinos form height is 150% that of their homid form height, with significantly more mass. Ironically, a Garou's dexterity increases in crinos form, when it should logically decrease. It's because they're part-spirit, probably.

It's a bit of a standing joke among BattleTech fans who have done the math that despite usually weighing anywhere from twenty to a hundred tons, given their physical dimensions BattleMechs must be less dense than water and should therefore logically float, especially since they're also by default environmentally sealed so they can operate in vacuum and other hostile environments. The rules, of course, still have them wading through any water deep enough to be worth depicting on the map. The square cube law exists In-Universe, but its magnitude is reduced, with the upper scale of feasible battlemech tonnage capped at 100 tons; you can build them larger ("superheavy"), but then they are barely able to sustain their own weight and move at a snail's pace even with an enormous reactor. Only three superheavy designs exist, and only two of those were functional designs — the 110 ton Matar's design team was executed for "treasonous incompetence" as the mech could barely move under its own power.

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In Ilivais X the Ilivais units are basically 80s super robots, and as such standing under their own weight would be impossible without assistance. This developed into "make them focused on flight" which then developed into "don't even bother giving them feet". As such, most of them have blade legs that end in a point, meaning that if they lose flight capability, they're utterly incapable of movement.

In the Whateley Universe story "Boston Brawl", there's an Author Tract explaining how size-warping 'giants' really work, since the Square-Cube Law and some other laws of basis physics would seem to make it impossible. The giant Matterhorn only appears to be a 40-foot giant because everyone else interfaces with his warp displacement field instead of him. The Workshop at Whateley contains a Humongous Mecha that devisor students occasionally work on. The best it's done is take three steps before the knee came apart. Played dead straight on occasion, too — Jimmy T's antics on Hallowe'en come to mind, as do any Shifter (as opposed to Warper) size-changers.

Discussed extensively in Small Problem.

The Salvation War averts this to a degree. On the one hand, most of the "lesser demons" are only about 8 feet tall, basically are made of muscles on top of more muscles, and while they all have wings only two subspecies (harpies and gorgons) can actually fly, it is stated that their bodies must produce lighter-than-air gasses to even manage that. On the other hand, there are many much larger demons, such as Satan, that well reach over 20 feet tall, although they tend to stay on the backs of great beasts or in their throne rooms. All Angels can fly, but they also are stated to have the gas producing organs. Yahweh is stated to be HUGE, but never moves from sitting on the Eternal Throne. When Heaven is conquered, the great gates to the Eternal City are so giant they cannot be moved, so must be blasted down.

Western Animation

On Chip 'n Dale Rescue Rangers, the main cast — two chipmunks, two mice, and a fly — are, encounters with size-changing rays aside, roughly the same size as their real-world counterparts, yet exhibit the same fear of falling a human should, given the same heights... often a distance that would likely only daze them for a few seconds. (This being Disney, however, something always lessens the threat to our heroes and their friends anyway, so whether their fears are founded or not is never shown.) A simpler example form the same show would be the cast frequently using large (for them) human-made tools with relative ease.

Lampshaded on The Venture Bros. with Humonguloid, a giant with a heart condition and other severe problems. Of course, VB's science of choice being superscience, he later gets shrunk to the size of an ant and survives for decades with no major health problems and, indeed, no medical care (Although he ends up being accidentally crushed to death by a rocking chair in the season 5 finale ). The same character states the "proportional strength of an ant" idea to likewise be nonsense.

). The same character states the "proportional strength of an ant" idea to likewise be nonsense. In Wakfu, this is how Sallygrove defeats Rubilax. Where every time Rubilax is hit, he grows in size and mass. However, them being in the middle of a desert, after getting too big he starts to sink into the ground. He chooses to return being trapped in a sword rather than dying suffocated in sand.

In Adventure Time, the Eldritch Abomination known as Orgalorg could move around freely — in outer space. When it was transported to Earth, gravity compacted and condensed its size so much, the creature took on a smaller form: a penguin .

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