A bacterium is about two thousandths of a millimetre long. A typical animal cell is ten or more times wider. A frog egg is several hundred times wider than the bacterium. Yet almost no cell on Earth is the size of a grape, let alone a fist. Something stops cells from simply growing. That something is geometry, and one sum explains it: how much skin a cell has compared with how much it has to feed.
Every cell takes in food and gets rid of waste through its outer membrane, its skin. The amount of skin goes up with the square of the cell's width, because skin is a surface. The amount of cell to feed goes up with the cube, because the inside is a volume. Make a cell twice as wide and it has 4 times the skin but 8 times the cytoplasm to supply.
For a sphere the ratio works out neatly. Area is 4πr² and volume is 4⁄3πr³, so skin per unit of volume is 3⁄r. Halve the radius and each bit of cytoplasm gets twice the skin. Double it and each bit gets half. A big cell is not just a bigger small cell. It is a worse-supplied one.
The disc below is one cell, always drawn the same size on screen. The slider sets how big it really is, from a bacterium up to a 2 cm ball. The dots are glucose molecules entering at the membrane and jiggling about at random. The lime band is how far they typically get in one second, so you can see which part of the cell is fed in that time and which part is still waiting.
Model: a solid ball of watery cytoplasm, glucose diffusion coefficient 600 µm²/s, time to the centre ≈ r²⁄6D. Real cytoplasm is more crowded, so real times are longer.
Try the presets. At bacterium and red blood cell sizes the whole cell is reached in a blink. At about 120 µm across, a second is just enough to reach the middle (the dashed ring appears just beyond that). By the frog egg, only a thin outer shell, about a quarter of the cell, is reached in a second, and the centre waits well over a minute. At 2 cm the wait is hours.
Inside a cell, molecules mostly do not travel in straight lines. They are jostled by their neighbours and wander. This is diffusion, and it has an awkward rule: the time it takes grows with the square of the distance. Ten times farther takes a hundred times longer. For glucose in water, with a diffusion coefficient of about 600 µm² per second, a trip of 10 µm takes around 28 milliseconds. A trip of a millimetre takes about 5 minutes (4.6, to be exact). A centimetre takes about 8 hours.
A cell that needs a steady supply cannot wait hours for fuel to reach its middle. So there is a ceiling, set partly by how fast the cell burns what it takes in. That is why most cells stay between a few micrometres and a few tens of micrometres across.
Nature has several tricks, and each one cheats the sum in a different way.
Be flat or thin. The red blood cell is about 8 µm across and, in humans, biconcave, a dimpled disc. No part of it is far from a surface. A nerve cell can be several metres long in a giraffe, but it is very narrow, so nothing inside is far from the membrane.
Fold the skin. Cells that absorb a lot, such as those lining the gut, crumple their membrane into tiny finger-like folds. The cell keeps its size but gains area.
Stay quiet. Eggs are the largest cells in many animals. A human egg is about 0.1 mm across, the biggest human cell by diameter, and a frog egg is 1.2 to 1.3 mm. Both are packed with stored supplies and spend most of their time waiting. The yolk of an ostrich egg is technically a single cell too, with its nuclear material in a small spot at the edge.
Hollow out the middle. In 2022 researchers described Thiomargarita magnifica, a bacterium from mangrove sediments in Guadeloupe. It is over a centimetre long, up to about two centimetres, and visible to the naked eye. Most of it is a fluid-filled vacuole, so the working cytoplasm is pushed into a thin layer around the edge. It is as if the cell stayed small by turning its inside into a balloon.
The same sum explains why cutting something up makes it react faster. Slice one big cell into 8 equal pieces and the total volume does not change, but each piece is half as wide, so the total skin doubles. That is the logic of dividing: small cells have the surface they need, and many small cells beat one big one. Next time you see a cell drawn as a tiny dot, remember it is not small by accident. It is as big as its skin can feed.