Answer key: Cell Size & SA:V Ratio Activity
For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student activity: Cell Size & SA:V Ratio Activity.
Goal: Calculate the surface area, volume and surface-area-to-volume ratio of model cells, and explain why most cells stay small.
Completed results table
| Cell | Surface area | Volume | SA : V ratio |
|---|---|---|---|
| Cube, s = 1 µm | 6 µm² | 1 µm³ | 6 per µm |
| Cube, s = 2 µm | 24 µm² | 8 µm³ | 3 per µm |
| Cube, s = 4 µm | 96 µm² | 64 µm³ | 1.5 per µm |
| Bacterium (sphere, r = 0.5 µm) | 3.142 µm² | 0.524 µm³ | 6 per µm |
| Animal cell (sphere, r = 10 µm) | 1,256.637 µm² | 4,188.79 µm³ | 0.3 per µm |
| Frog egg (sphere, r = 500 µm) | 3,141,592.654 µm² | 523,598,775.598 µm³ | 0.006 per µm |
Answers
What happened to the SA : V ratio each time you doubled the side of the cube? Use SA : V = 6 / s to explain.
It halved each time: 6, then 3, then 1.5 per µm. Surface area grows with s² but volume grows with s³, so the ratio 6s² ÷ s³ = 6/s falls as s grows.
Calculate the surface area, volume and SA : V ratio of a spherical cell with a radius of 10 µm. Show your working.
SA = 4πr² = 4 × π × 10² = 1,256.6 µm². V = 4⁄3 πr³ = 4⁄3 × π × 10³ = 4,188.8 µm³. SA : V = 3/r = 0.3 per µm.
The bacterium has an SA : V ratio of 6 per µm and the frog egg 0.006 per µm. How many times larger is the bacterium’s ratio, and what does that mean for supplying each cell by diffusion?
6 ÷ 0.006 = 1,000 times larger. Each cubic micrometer of the bacterium has 1,000 times more membrane for exchange, so diffusion supplies it easily. The center of the frog egg is 500 µm from the membrane, so the egg relies on its stored yolk and divides into many smaller cells as the embryo develops.
Predict what happens to a cell’s SA : V ratio if it doubles in diameter, then check with the SA : V if twice as large readout.
It halves. For the 20 µm animal cell the ratio drops from 0.3 to 0.15 per µm, because doubling every length multiplies surface area by 4 but volume by 8.
Give two ways that real cells increase their surface area without becoming smaller, with an example of each.
Folding or extending the membrane, such as the microvilli on intestinal cells or root hairs on plant root cells. Taking a long, thin or flattened shape, such as neurons or the biconcave discs of red blood cells.
Challenge: a cube-shaped cell has an SA : V ratio of 0.12 per µm. How long is its side?
SA : V = 6/s, so s = 6 ÷ 0.12 = 50 µm. The “Plant cell (50 µm across)” preset confirms 0.12 per µm.