# Small is Mighty: the Square-Cube Law

What do you think is stronger,
an ant or an elephant? An elephant can lift
more weight than an ant, but what about their strength
compared to their body size? The answer might surprise you. An elephant can lift
500 kilograms, which is 10% of its body weight. An ant can only live
one-millionth as much, but that amount is actually
50 times its body weight. Ants use their
impressive strength to move food and materials
to their colonies. How much can you lift? The average person between the
size of an ant and an elephant can only lift about
half of its own weight. Why is an ant so strong? First, we need to understand
a principle called the square cube law,
which Galileo came up with over 400 years ago. He observed that when
an object grows in size, its volume increases
faster than its area. Let’s picture a cube. The area of one face on the
cube is a length squared, and the volume is a
length cubed, hence, the square cube law. This cube has sides that
are 1 centimeter long. The area of a face on the
cube is 1 squared centimeter, and its volume is
1 cubic centimeter. Now this cube has sides
that are twice as long. The area of its face is
4 squared centimeters, but the volume is now
8 cubic centimeters. Doubling the length of the
sides quadruples the area, but octuples the volume. As the object gets bigger,
the ratio between its area and volume changes. They’re important consequences
when an objects volume grows faster than its area,
because it changes the way, it could interact
with its surroundings. The effects of the
square cube law can be observed in
our everyday life, for example, in
designing buildings. We can make small scale
models of buildings out of paper mache,
but real houses are made of wood
brick and metal. But why can’t we just build a
real house out of paper mache? The amount of weight that
a structure can support is relative to the
cross-sectional area, but the buildings own weight
is proportional to volume. Since the square cube law told
us that volume increases faster than area, a bigger
building will weigh more, but might not have enough area
to support its own weight. These two towers are made
of the same materials, but only the small one can
support its own weight. Because of this engineers
need to use stronger materials like bricks to design
real buildings that have larger dimensions. For a similar reason,
an ant can carry more, relative to its body
weight, than an elephant. And this is also the reason
why monster ants do not exist. Strength is proportional
to a cross-sectional area of their legs, while weight is
proportional to their value. Here you see a small
ant holding its weight. If we now scale
up the ant, we see that the ant’s legs can no
longer support its weight, and it collapses. And hey look, the small
ant can even bring treats back to its friends. The square cube law
is also observed in aerodynamics, which is the
science of flying and falling. You may have learned that
gravity causes matter to fall towards the
ground, and that gravity makes objects of any mass
fall at the same speed. Why is it then,
that in real life, some objects seem to
fall faster than others? A falling object actually
experiences two forces, gravity and air resistance,
which is also called drag. Gravity makes the
objects falls faster, while drag slows it down. The force from gravity is
proportional to the volume of an object, while drag is
proportional to its exposed area. Remember again that
when things get bigger, volume increases
faster than area. If we drop these two different
size helicopters from four stories high, what happens? As predicted, the
bigger helicopter fell faster, because the
drag, which goes with area, increases more slowly than
weight, which goes with volume. Let’s review what
we’ve learned today. The square cube law
tells us that when an object grows in size,
its volume increases faster than its area. We’ve seen how this can
be applied to buildings, animals, and falling objects. Look around you,
what else can you explain with the
square cube law?

## 46 thoughts on “Small is Mighty: the Square-Cube Law”

• ### Sophia M Post author

First Comment YAY!!!! 😀

Congrats 🙂

• ### rasecsexi Post author

I learn something today. It was Awesome! d=(^.^)z

• ### Chiranjeev Thomas Post author

strength–directly proportional– to cross section area of the legs??? ……. doesn't an opposing force like the muscles or the MATERIAL used to support the structure matter?? i mean.. one can balance an insane amount of weight on 4 diamonds standing as pillars but covering a small surface area

2:46?

Thx

• ### Derkasaurus Post author

"average human can only lift half his own weight" looks like somebody has never been to the gym.

• ### Melanie Hevel-Olson Post author

@NevolmonGaming *SMH*… I hope you were kidding with that ignorant comment and that you do actually understand the meaning of "average" when trying to generalize to billions of people. If you weren't kidding, I'd be willing to explain it to you 😛 I'm concerned you weren't kidding simply because I cannot imagine making such an inane statement in response to a video like this.

• ### haseothepkker Post author

I WANT THOSE STAR BURSTS!!
I WANT THEM THEY'RE MINE!!

• ### Ichigo Bleach Post author

Its only my humble opinion.

All of this is only HALF TRUE. Gravity play a MAJOR ROLE. Unless u ENLARGE an ANT, maybe ANT also cant SUPPORT more weight because of GRAVITY.

Also CUBE LAW sometimes doesnt apply to ALL. There is sometimes LARGER object with LARGER volume but with SMALLER MASS. I mean THE CUBE is LIGHT. Compare to SMALLER CUBE.

• ### Angel Alvarez Post author

I have a friend in high school who is only 163 pounds and can lift almost 400 pounds.

• ### Adam Howlett Post author

"Gravity makes objects at any mass fall with the same speed"

Oh dear.

• ### carultch Post author

An elephant is 100 times larger than a mouse per linear size, and weighs 500000 times as much.  Lighter than equal densities/equal proportions would predict, but it is still the right order of magnitude.

A mouse's femur is 1.1 cm long, with a diameter of 0.7 mm.
An elephant's femur is 1 meter long.

If direct compression pressure were the governing failure mode, we'd expect an elephant's femur to have a diameter of 50 cm.  A square-cube law, exactly as this video would predict.  This is an extremely impractical femur, to be so thick that its thickness is half its length.

If buckling were the governing failure mode, we'd expect an elephant's femur to have a diameter of 18 cm.  This seems much more reasonable, but it still is seems rather thick for a bone proportion.

The actual femur of an elephant is 8.6 cm in diameter.  The ratio of diameter to length is only slightly larger for the elephant (0.086) than for a mouse (0.063).  Not nearly enough to justify either scaling argument.

Something else is going on here.  Anyone know what it could be?

The elephant

• ### Abdallah Ezzat - Koji Post author

the square cube law should work all the time right.

but it confuses me when we deal with fractions.

lets say we have a cube, its volume is L^3, and its cross section is L^2.

So if we start with L=0.1("unit"), then its cross section is 0.01 (unit squared), and the volume is 0.001 (unit cubed).

When the dimension doubles to become 0.2 (unit), the cross section grows four times which is 0.04(unit squared), and the volume grows eight times faster which is 0.008 (unit cubed).

Hummmm…..

I knew what what seemed wrong, I mistook that the values before 1 (unit) is greater for lower powers. However, still the higher powers always have a higher rate of increasing (which is the idea), and the values pass the breaking point at 1 (unit).

It creates an illusion jumping between different units of measure. take a 2 cm, the cross section is 4 cm^2, and the volume is 8 cm^3, it is not right to compare different types of quantities e.g. a volume and a cross section. still numerically 8 is greater than 4, "the volume is greater than the cross section". If we take the same cube and measure it in meters, its length is 0.02 m, cross section 0.0004 m^2, and the volume 0.000008 m^3. numerically 0.000008 is smaller that 0.0004, "the volume is smaller than the cross section".

Anyways it is one of the times when you know the answer to your question right after you speak out you question.

• ### Johan Westin Post author

1:05 [Gallileo Intensifies]

• ### KeppaKop Post author

0:48 do you even lift bro?

• ### Marco Polo Post author

This. Is a pile of shit atoms. are round not square so this theory. Is garbage just look at the ant. Its body part are round not square infact a small square. I could smash easy with a hammer. If. Its steel but a large steel square would take me hour's to smash. what century are you living. In. If. It was a small steel ball. It would be harder for me even. If. It was bigger. It would still be hard to smash

• ### Marco Polo Post author

and. If square's are weaker when there bigger why are building square why. In the hell would anyone make a square house or building period this shit. Is stupid

• ### Marco Polo Post author

and the ant. Is able to pick more weight because gravity dont effect smaller objects. that's why the ant has a shell around. It's body because gravity has almost no effect on. It so. It has pure muscle beneath the shell

• ### Richard Strawn Post author

god your voices are so fucking annoying.

• ### blessvic Post author

what are the example of economic of scale due to cube square rule

• ### Rulersonic Boom Post author

Hmm Good explanation here's a sub for ya.
Clicks the subscribe button

• ### Nate Baker Post author

Where is the actual equation?

• ### TheKKKandTerrorist Post author

elephants can lift 9000-10000kg wtf 300-500kg is just their trunk alone….

• ### Goat Post author

HALF OF MY WEIGHT?! Bitch I weigh 130 and squat 160. What you talking about 'half of his own weight'
Weak ass plebs

• ### Aaron Dray Post author

This is why honey I shrunk the kids could not happen.

• ### Reese Archer Post author

so if you have a bunch of ants working together to make one big ant-elephant, would the ant-elephant be stronger than the elephant?

2:16– Rain

• ### dazhibernian Post author

The ants are on roids.

• ### Sealson Kataru Post author

This is a great video

• ### Slxy-Faulty Post author

im 14 and I weigh 75 kg and lift 80

• ### superzilla784 Post author

Funny thing is that the Square Cube Law was actually mentioned in the 2016 movie. when Godzilla first arrived at Japan, the senator called a meeting to calm everyone down. "The creature cannot come up onto land without crushing itself under its own weight" but then Godzilla gave a middle finger to science and came up onto land anyway. Lol

• ### Sheev Palpatine Post author

So most people cannot do pullups?

• ### Mac man Post author

..In this video: https://www.youtube.com/watch?v=WQGw1tKEpEg
The Antichrist Irrefutably Identified! • The “Beast” …666

• ### Dr Ankit Panigrahi India Post author

You ignored a crucial parameter and that's density .

• ### Tomas Drake Post author

LOBO

According to the science present here, I'm not actually that fat.

• ### Hacher Unfriended Post author

An ant can lift 50x its weight but if it's weight is twice bigger, it topples. What's logic in that?

• ### Kevin Mario Post author

Who else came here from Yahtzee's RE2 review?

• ### Menace Vang Post author

Damn Galileo be making everyone look like a slacker

• ### Milk and Cookies Productions Post author

Who else watched cuz they though it was a dick joke?

• ### Kieran m Post author

Not even gonna lie, just thought of attack on titan the whole time

• ### Shammo Hamid Post author

I love physics.

• ### tommy d u b b s Post author

This is like how something is faster the smaller it is….but this is about strength

• ### Seth Gizen Post author

Well time to shrink myself to the size of a quark.