Surface area to volume ratio

The geometry that keeps cells small.

The idea

A cell takes in oxygen and nutrients and expels waste across its surface, but the demand for those exchanges comes from its whole volume. For a cube of side s the surface area is 6s² and the volume s³, so the ratio SA : V = 6/s — for a sphere it is 3/r — which means the ratio falls as the cell grows. Fick’s law says the rate of diffusion is proportional to surface area times concentration gradient divided by distance, so a large cell has too little surface for its volume and too far for molecules to travel to its center; most cells therefore stay between 10 and 100 µm. Living things get around the limit by dividing into many cells, by flattening or elongating, by folding membranes (microvilli in the gut, cristae in mitochondria), and, in large animals, by building dedicated exchange surfaces such as the roughly 70 m² of alveoli in a pair of human lungs.

Work through an example

A 1 µm cube has surface area 6 µm² and volume 1 µm³, so SA : V = 6 : 1. A 10 µm cube has surface area 600 µm² and volume 1,000 µm³, so SA : V = 0.6 : 1 — ten times less surface per unit of volume. At 100 µm the ratio drops to 0.06. Now split that 10 µm cube into 1,000 cubes of 1 µm: the total volume is still 1,000 µm³, but the total surface is 6,000 µm², ten times more. That is the whole reason an elephant is made of trillions of small cells rather than one enormous one.

What to watch for

Bigger cells do have more surface area — the mistake is assuming that helps. When a length doubles, surface area grows by 2² = 4 times but volume by 2³ = 8 times, so surface per unit volume is cut in half. A related myth is that large animals have large cells: an elephant’s cells are about the same size as a mouse’s; the elephant simply has more of them.