Surface area and volume.

Step 1 of 6

Skin

Everything a cell or an animal needs, oxygen, food and water, comes in through its surface: the cell membrane, the skin, the lungs, the gut wall. And whatever it uses, it uses throughout its volume. That is why the ratio between the two matters so much: surface area to volume.

The Scene shows a solid-oak kitchen table beside a copy of it 3 times bigger in every length: height, length and width. Slice and count is on Slices and cubes: every leg is sawn through, and a cube is lifted out of each top. The copy’s leg slice is gridded in squares the size of the original’s slice.

Step 2 of 6

Body

Now look at the cubes lifted out of the tops: the copy’s cube is gridded in cubes the size of the original’s.

Step 3 of 6

The ratio

Area grows as $k^2$⁠ and volume as $k^3$⁠, so volume always wins. The bigger the animal, the fewer square centimetres of skin, lung and gut it has for each cubic centimetre.

Step 4 of 6

A quarter

Size is free now, starting at the original’s size.

Step 5 of 6

Cubes

The same sum works for any shape, but it is easiest to see in a cube. A cube has six faces, each a square.

Step 6 of 6

Why cells are small

Shrink the table to toy size and the ratio grows: a lot of surface for its volume. An animal is the same: a rat has far more skin for each cubic centimetre than an elephant. And a cell is thousands of times smaller than a rat. A 10 µm cell gets enough oxygen for all its volume through its membrane by diffusion alone, and the way from its membrane to its middle is short.