Why don't cells just grow bigger? The answer is geometry. You know how to find surface area and volume. A cell takes in food and gets rid of wastes through its surface, but its whole volume needs them.
Key idea
When a cell gets wider, its volume grows faster than its surface. Double the width: the surface area grows 4 times, but the volume grows 8 times. So a large cell has too little surface for its inside, and small cells exchange materials faster.
Why it works
Surface area uses side × side, so it grows with the square of the width. Volume uses side × side × side, so it grows with the cube. For a cube, surface area ÷ volume $= 6s^2 \div s^3 = 6 \div s$: the bigger $s$ gets, the smaller this ratio.
Worked examples
Worked example: Find the ratio
A cube-shaped cell has sides of 2 µm. Find its surface area, its volume and the ratio.
- Surface area: $6 \times 2 \times 2 = 24$ µm².
- Volume: $2 \times 2 \times 2 = 8$ µm³.
- Ratio: $24 \div 8 = 3$.
Answer: 24 µm², 8 µm³, ratio 3 : 1.
Worked example: Doubling the width
The cell grows to 4 µm on each side. How do its surface and volume change?
- Surface area: $6 \times 4 \times 4 = 96$ µm², which is $96 \div 24 = 4$ times as much.
- Volume: $4 \times 4 \times 4 = 64$ µm³, which is $64 \div 8 = 8$ times as much.
- Ratio: $96 \div 64 = 1.5$, half of what it was.
Answer: Surface ×4, volume ×8, ratio cut in half.
Common mistake
Thinking a cell that doubles in width doubles its surface and volume too. Surface grows 4 times and volume 8 times. Check with small cubes: a 2 × 2 × 2 cube is built from 8 unit cubes.