Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide.
Recap
- Everything a cell takes in or gets rid of, including heat, crosses its surface. Every part of its volume creates demand.
- As a cell grows, volume rises faster than surface area, so SA/V falls and demand per unit of surface rises. This limits cell size and shape.
- Smaller cells have a higher SA/V and exchange materials more efficiently.
- Flat or long shapes raise SA/V compared with a sphere of the same volume.
- Membrane folds and projections add area for almost no volume: microvilli (gut lining), root hairs, cristae; leaves exchange gases through stomata controlled by guard cells.
- Larger organisms have a lower SA/V, so they exchange heat more slowly per unit mass. Smaller organisms usually have a higher metabolic rate per unit body mass.
Key relationships
| Shape | SA | V | SA/V |
|---|---|---|---|
| Cube | 6s² | s³ | 6/s |
| Sphere | 4πr² | (4/3)πr³ | 3/r |
| Rectangular solid | 2lh + 2lw + 2wh | lwh | calculate |
| Cylinder | 2πrh + 2πr² | πr²h | calculate |
| Change | Effect |
|---|---|
| length × k | SA × k², V × k³, SA/V ÷ k |
| supply = SA × (uptake per unit area) | demand = V × (use per unit volume) |
| supply = demand | SA/V = demand rate ÷ supply rate |
Assumptions behind the numbers
- Cells are modelled as simple shapes; real cells are irregular.
- Supply-and-demand models assume uptake is the same everywhere on the membrane and ignore distances inside the cell.
- SA/V has units (µm⁻¹ when lengths are in µm); only compare ratios in the same units.
Mistakes to avoid
- Saying a larger cell has less surface area (it has less per unit volume).
- Leaving out the two ends of a cylinder (2πr²).
- Using diameter in place of radius.
- Dropping the units of SA/V.
- Thinking microvilli help by enlarging the cell rather than adding surface.
- Reversing the metabolic-rate trend: small organisms have the higher rate per unit mass.
Quick self-check
- SA/V of a cube of side 5 µm? (6 ÷ 5 = 1.2 µm⁻¹)
- SA/V of a sphere of radius 6 µm? (3 ÷ 6 = 0.5 µm⁻¹)
- Why does a mouse need more food per gram of body than a horse? (Its higher SA/V means it loses heat faster per gram, so it needs a higher metabolic rate per unit mass to stay warm.)
Next: practice questions.