The Square-Cube Law: Why Giants Can't Just Be Scaled Up

0:00Could a cat made 10 times bigger still stand on its legs?

0:05A scale factor of 10 means every length is multiplied by 10.

0:11Model the animal as a cube, 1 centimetre on each side, scaled up to 10 centimetres.

0:19Bone strength depends on cross-section: the area of a cut straight across, like this slice.

0:27The big cross-section is 10 by 10, or 100 square centimetres, so it is 100 times as strong.

0:36Weight depends on volume, the space it fills.

0:39The big cube holds 10 layers of 100 small cubes: 1000 cubic centimetres, so 1000 times as heavy.

0:49So strength grows 100 times, but weight grows 1000 times.

0:55Load per square centimetre is weight divided by cross-section: 1000 divided by 100 is 10 times as much.

1:05Check: each square of the big cross-section holds up a column of 10 cubes, not just 1.

1:13This is the square-cube law: area grows by 10 squared, or 100, and volume by 10 cubed, or 1000.

1:23A cat scaled up 10 times puts 10 times the load on each square centimetre of bone, so its legs could break.

1:34That is why an elephant has much thicker legs for its size than a cat.

1:39A common mistake is expecting a design that works small to work the same way at full size.

1:45Scale up 10 times: strength grows 100 times, weight 1000 times, so each square centimetre carries 10 times the load.