Godzilla and the Square-Cube Law: How Big Can Animals Get?

A film monster has to answer two physics questions before it answers the food one. Can its legs carry it? Can it shed the heat its body makes? Both come from one piece of geometry that Galileo set down in the seventeenth century. This page works the geometry through with real numbers, then uses it to estimate what Godzilla would eat if he could exist at all.

What is the square-cube law?

When an object grows by a factor k in every direction, its surface area grows by k² and its volume by k³⁶. Galileo first described the relationship in 1638 in Two New Sciences⁶. The result is that volumes change faster than areas, and areas change faster than linear dimensions⁵.

A cube with sides of 1 meter has 6 square meters of surface and 1 cubic meter of volume⁶. Double the sides and the surface becomes 24 square meters, while the volume becomes 8 cubic meters⁶.

Worked example (our arithmetic): what growing by a factor k does to a shape

Keep the last column in mind, because it decides two things. Strength depends on area, and weight depends on volume. A bigger body has less surface for each kilogram it has to hold up, cool and feed.

Does the square-cube law apply to animals?

Yes. The force a muscle can produce and the strength of a bone are proportional to their cross-sectional areas, while the weight of an animal is proportional to its volume⁵. Other jobs split the same way: oxygen uptake follows the area of the lungs, digestion the area of the gut and heat loss the area of the body, but the oxygen and food that must be supplied and the heat that is produced follow mass⁵. If an animal works well at one size, a change of size alone puts these functions out of step⁵.

Bones

If an animal grows with no change in shape, the load on its bones rises with volume and the bone’s cross-section only with area⁵. Eventually the bones break under the animal’s own weight⁵. Real animals avoid that by changing shape: larger animals have proportionally stouter, thicker bones⁵, and an elephant’s bones are proportionately much larger than a mouse’s⁶.

J. B. S. Haldane drew the lesson for giants in his 1928 essay On Being the Right Size⁶. A square inch of a giant’s bone had to support 10 times the weight of a square inch of human bone, and the human thigh bone breaks at about 10 times body weight⁶.

Measured bone stress shows how tight the margin is. Bone breaks at about 200 megapascals, and the working stress in walking mammals runs about a fourth to a third of that⁵. The safety factor is only 3 to 5, and it stays constant from chipmunks to elephants, a 25,000-fold range, because bone shape and posture change⁵. An elephant’s legs are stacked in an almost vertical position under its body, which keeps the load directed along the bone¹⁵.

Heat

Large mammals such as elephants have a harder time cooling themselves than small ones such as mice⁶. Elephants use their ears: blood in the ear veins is cooled as the ears flap, and body temperature falls by several degrees¹.

Energy

Surface-area reasoning suggested that metabolic rate should scale with mass to the power 2/3⁹. Kleiber found that the basal metabolic rate fits a power of 3/4 better³⁹. It still lacks a theoretical explanation that is entirely satisfactory³. That matters for the food question below.

Does Godzilla violate the square-cube law?

Yes, if he is a scaled-up animal. The 1954 film scaled Godzilla to 50 m tall so that he could just peer over the largest buildings in Tokyo². Character profiles list his weight as 20,000 to 60,000 metric tons². Adult Asian elephants weigh 6,000 to 12,000 pounds, or 2,750 to 5,420 kilograms, and stand 1.8 to 3.8 meters at the shoulder¹.

Worked example (our arithmetic): the biggest elephant scaled to Godzilla’s height

The scaled mass of 12,300 metric tons is the same order as the 20,000 to 60,000 tons in the profiles², so the elephant comparison is fair. The same arithmetic is in LaBarbera’s lecture for King Kong: a fivefold increase in height means a 25-fold increase in bone cross-section, a 125-fold increase in mass and about five times the stress⁵. Godzilla’s factor is 13.2, not 5.

Two repairs are available in biology. The first is stouter bones, as in the scaling from mouse to elephant⁵⁶. The second is posture, since straight legs keep loads along the bone⁵. Both together hold the safety factor for animals up to elephant size⁵. Elephants already use both repairs, so a further 13.2 times on the stress has nowhere to come from (our arithmetic)⁵.

Heat is the second problem. The area for each unit of volume falls to 1 ÷ 13.2, which is 0.076 of the elephant’s, so each kilogram of Godzilla has about 7.6 percent as much skin to lose heat through (our arithmetic)⁶¹². The body heats faster than it cools. Bodies lose heat through their surface but produce it throughout their mass³.

Do dinosaurs violate the square-cube law?

No. The sauropods were the largest land animals ever, surpassing the largest herbivorous mammals by an order of magnitude in body mass⁷. Several lineages produced giants above 50 metric tonnes by conservative estimates⁷. Published estimates for the titanosaur Argentinosaurus run 70 to 90 tonnes⁷. Most other dinosaur lineages rarely exceeded 10 tonnes⁷.

They stayed inside the law by changing their build. Sauropod skeletons combined dense limb bones as thick as tree trunks, vertebrae so riddled with air sacs that they resemble honeycomb, and small skulls of sometimes paper-thin bone¹⁰. The limb bones work like columns under a building, and the weight a column carries grows with its thickness¹⁰. An erect, column-like forelimb evolved in parallel, and that was necessary for sauropods to reach multi-tonne mass⁷.

The air sacs lowered weight for the volume: living sauropods probably had a specific density of about 0.8 kilograms per liter, closer to a bird than to a mammal⁷. An avian-style respiratory system would also have cut the cost of breathing, reduced specific gravity and helped remove excess body heat⁷. Heat and weight, the two problems above, were both addressed.

Worked example (our arithmetic): Godzilla against the biggest real dinosaur

Even the lightest profile weight is more than 200 times beyond the biggest land animal known⁷². A dinosaur is a calibration point, and Godzilla is far past it.

What is the biggest an animal could theoretically get?

On land, the theoretical limit is above 100 tonnes. Research by Jyrki Hokkanen at the University of Helsinki suggests that the biomechanical limit for terrestrial body mass is well over 100 metric tons¹⁰. The limit comes from a trade-off: thicker limb bones carry more weight, but mobility decreases as they thicken, so there is an upper limit on thickness for a functional animal¹⁰. Published estimates for the largest sauropod, 70 to 90 tonnes, sit under that ceiling⁷.

In water the rule bends, because the buoyancy of water offsets some of the effects of gravity⁶. That is the primary reason the largest animals ever to exist are aquatic⁶. The blue whale is the largest animal ever to live on our planet⁸. Antarctic blue whales reach up to about 110 feet and more than 330,000 pounds, while North Atlantic and North Pacific whales grow to about 90 feet and over 100,000 pounds⁸. A sauropod standing in water would struggle to breathe if its lungs were deeper than 8 to 10 feet, so standing in a lake does not rescue a land giant⁵.

How much would Godzilla eat?

Food is the slowest part of the answer, because it needs Kleiber’s law. For an animal of mass w kilograms, the basal metabolic rate is about 70 × w^0.75 kilocalories a day³. The exponent is the key. A cat 100 times the mass of a mouse uses only about 32 times the energy, which is 100^0.75³.

Worked example (our arithmetic): resting energy by Kleiber’s law

The person’s 1,694 sits close to the 2,000 calories a day used as a general guide for nutrition advice⁴. Godzilla’s low-end 20.9 million is 20.9 million ÷ 2,000 = about 10,467 times that guide, or roughly 10,500 people’s daily food (our arithmetic)³⁴².

The exponent saves him a great deal. If appetite grew in direct proportion to mass, Godzilla at 20 million kg would weigh 20 million ÷ 70 = 285,714 times a person and would need 285,714 × 1,694 = about 484 million kcal a day (our arithmetic)³. Kleiber’s law gives 20.9 million, which is 23 times less³. The exponent works in reverse too: a shrunken person in a classic film would have to eat his own weight daily, like a shrew⁵.

In tons of food

An elephant is the closest real animal for a food estimate, and it eats about 165 to 330 pounds a day, assimilating only about 44 percent of that¹. Godzilla’s mass is 20 million ÷ 5,420 = about 3,690 times a top-size elephant’s, and 3,690^0.75 is about 473 (our arithmetic)³¹². So his resting needs are about 473 elephant-days of food each day. At 165 to 330 pounds an elephant-day, that is 78,000 to 156,000 pounds, or about 39 to 78 tons at 2,000 pounds a ton (our arithmetic)¹. The estimate assumes Godzilla eats the same food as an elephant, and a diet of denser food would weigh less.

Treat every number here as an estimate. Kleiber’s law describes basal rates, not a rampage, and both the 3/4 exponent and its explanation are debated³⁹. Read the totals as one estimate, not a measurement.

Why does Godzilla keep getting bigger?

Godzilla has doubled in size since 1954, a pace far beyond the rate of evolution seen in 2,500 natural organisms¹¹. The 2014 American film scaled him to 108.2 m and 81,650 metric tons². Against the 1954 figures that is 108.2 ÷ 50 = 2.16 times taller, and a square-cube scale-up of that height would multiply mass by 2.16³ = about 10 (our arithmetic)². The 2014 mass is only 81,650 ÷ 20,000 = about 4.1 times the 1954 low end, so the 2014 design is lighter than a straight scale-up of the 1954 one (our arithmetic)².

What to take from the numbers

Size never scales one thing at a time. Weight grows as the cube, strength and cooling as the square, and food as a power in between. For a smaller case of the same effect, read how fast coffee cools, and for loads that add up across an area, see the weight of snow. Both sit in the World section, next to the high-voltage page.

The best sources behind this page are the University of Chicago biology lecture, the sauropod review in Biological Reviews and the NOAA blue whale profile.