AP® Biology review sheet from Aim for Five (aimforfive.com/bio/units/2/2-2)
Unit 2 · Topic 2.2
2.2 Cell Size
A cell exchanges materials with its surroundings only through its surface, but its needs grow with its volume. Because volume grows faster than surface area, the surface area-to-volume ratio drops as a cell gets bigger. That limit explains why cells are small, why many have folds and projections, and why small animals lose heat faster than large ones.
Key terms
- surface area
- volume
- surface area-to-volume ratio
- membrane folds
- metabolic rate
Surface area vs. volume
Everything a cell takes in (oxygen, nutrients, water) and everything it gets rid of (wastes, heat) has to cross the plasma membrane. So the surface area of the membrane sets how fast exchange can happen. The volume of the cell sets how much material the cell needs and how much waste it makes.
As a cell grows, its volume increases faster than its surface area. For a cube with side length s, surface area is 6s² but volume is s³. Doubling the side multiplies surface area by 4 but volume by 8. So the surface area-to-volume ratio (SA:V) goes down as the cell gets bigger.
A small SA:V means a lot of cell interior for each bit of membrane. Past a certain size, the membrane can't move materials in and out fast enough to keep up, and the cell can't survive. That's a main reason cells stay small, and why big organisms are made of many small cells rather than one giant one.
Formulas you'll use
These formulas are on the AP Biology formula sheet. Make sure the units match: if lengths are in micrometers (µm), surface area is in µm², volume in µm³, and SA:V in µm⁻¹.
| Shape | Surface area | Volume |
|---|---|---|
| Cube (side s) | SA = 6s² | V = s³ |
| Rectangular solid (l, w, h) | SA = 2lh + 2lw + 2wh | V = lwh |
| Sphere (radius r) | SA = 4πr² | V = (4/3)πr³ |
| Cylinder (radius r, height h) | SA = 2πrh + 2πr² | V = πr²h |
Ways to boost surface area
When cells or organs need a lot of exchange, their shapes add surface area without adding much volume. Long, thin or flattened shapes have a higher SA:V than a cube or sphere of the same volume. Folds and projections do the same thing.
- Microvilli: tiny finger-like projections on the cells lining your small intestine (gut epithelial cells) multiply the area for absorbing nutrients.
- Root hairs: long, thin extensions of root cells increase the area for absorbing water and minerals.
- Membrane folds inside cells: the cristae of mitochondria and the thylakoids of chloroplasts pack more membrane, and the proteins on it, into a small space.
- You may also see cilia, and the stomata and guard cells of leaves, used as examples of structures that affect how much exchange happens across a surface.
Size, heat and metabolism in whole organisms
The same geometry works for whole animals. A small animal has a large surface area relative to its mass, so it exchanges heat with its environment quickly. A large animal has a small SA:V, so it gains or loses heat slowly.
To stay warm, small endotherms (animals that make their own body heat, such as birds and mammals) must make heat fast. So, in general, the smaller the organism, the higher its metabolic rate per gram of body mass. A shrew burns far more energy per gram than an elephant does and must eat almost constantly.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Comparing SA:V for cubes
Three cube-shaped cells have side lengths of 1 µm, 2 µm and 4 µm. Calculate the SA:V ratio of each, and explain which cell exchanges materials most efficiently.
Show the solutionHide the solution
- Step 1: Use SA = 6s² and V = s³ from the formula sheet.
- Step 2: 1 µm cube: SA = 6(1)² = 6 µm²; V = 1³ = 1 µm³; SA:V = 6/1 = 6 µm⁻¹.
- Step 3: 2 µm cube: SA = 6(2)² = 24 µm²; V = 2³ = 8 µm³; SA:V = 24/8 = 3 µm⁻¹.
- Step 4: 4 µm cube: SA = 6(4)² = 96 µm²; V = 4³ = 64 µm³; SA:V = 96/64 = 1.5 µm⁻¹.
- Step 5: Each doubling of side length halves the SA:V ratio. The 1 µm cell has the most membrane per unit of volume to serve.
Answer: SA:V = 6, 3 and 1.5 µm⁻¹. The 1 µm cell exchanges materials most efficiently because it has the highest SA:V.
- Example 2Calculator allowed
Agar cube diffusion
Agar cubes with sides of 1 cm, 2 cm and 3 cm are soaked in a dye for 10 minutes. In that time, dye diffuses 0.5 cm in from every face. What percentage of each cube's volume does the dye reach?
Show the solutionHide the solution
- Step 1: Find the part the dye doesn't reach: it's a smaller cube in the center. Its side is the original side minus 0.5 cm from each of two opposite faces, so side − 1 cm.
- Step 2: 1 cm cube: inner side = 0 cm, so nothing is unreached. Dye reaches 100%.
- Step 3: 2 cm cube: total V = 8 cm³; inner cube side = 1 cm, V = 1 cm³. Reached = (8 − 1)/8 = 7/8 = 87.5%.
- Step 4: 3 cm cube: total V = 27 cm³; inner cube side = 2 cm, V = 8 cm³. Reached = (27 − 8)/27 = 19/27 ≈ 70.4%.
- Step 5: Interpret: the larger the cube, the smaller the share of its volume that diffusion can supply in the same time. This models why large cells would starve in the middle.
Answer: 100%, 87.5% and about 70.4% of the volume for the 1, 2 and 3 cm cubes.
- Example 3Calculator allowed
Same volume, different shape (classic trap)
Cell A is a cube 2 µm on each side. Cell B is a long rectangular cell 1 µm × 1 µm × 8 µm. A student says they exchange materials equally well because they have the same volume. Is the student right?
Show the solutionHide the solution
- Step 1: Check volumes: A = 2³ = 8 µm³; B = 1 × 1 × 8 = 8 µm³. They are equal.
- Step 2: The trap: equal volume doesn't mean equal surface area. Calculate SA for each.
- Step 3: A: SA = 6(2)² = 24 µm², so SA:V = 24/8 = 3 µm⁻¹.
- Step 4: B: SA = 2lh + 2lw + 2wh = 2(1)(8) + 2(1)(1) + 2(1)(8) = 16 + 2 + 16 = 34 µm², so SA:V = 34/8 = 4.25 µm⁻¹.
- Step 5: B has more membrane for the same volume, so it exchanges materials faster.
Answer: No. The long, thin cell has SA:V = 4.25 µm⁻¹ vs. 3 µm⁻¹ for the cube, so it exchanges materials more efficiently.
Common mistakes
- Saying bigger cells have more surface area, so they exchange materials better. Total surface area does grow, but SA:V shrinks, and SA:V is what limits exchange.
- Dividing volume by surface area instead of surface area by volume. Write SA:V = SA ÷ V.
- Dropping or mixing units. A ratio of area to volume has units of 1/length, such as µm⁻¹.
- Saying small animals have lower metabolic rates because they're small. Per gram of body mass, smaller animals usually have higher metabolic rates.
On the exam
- Expect a calculation with formula-sheet shapes, followed by an 'explain' part. Points come from linking a higher SA:V to faster exchange of materials or heat.
- When a question shows a structure like microvilli, root hairs or cristae, say that it increases surface area relative to volume and name what is being exchanged.
Connected topics
Videos
Check yourself
4 questions on 2.2 Cell Size. Pick an answer to see if you got it, and why.
| Cube side length (cm) | Surface area (cm²) | Volume (cm³) | Percent of volume that changed color after 10 min |
|---|---|---|---|
| 1 | 6 | 1 | 87.5 |
| 2 | 24 | 8 | 57.8 |
| 3 | 54 | 27 | 42.1 |
Experimental data: cubes of pink agar containing an indicator were soaked in a clear acid solution for 10 minutes. The indicator turns clear as acid diffuses in.
What is the surface area-to-volume ratio of the 3 cm cube?
Which of the following claims is best supported by the data?
Shrews are very small mammals that must eat almost constantly. Based on surface area-to-volume reasoning, which of the following best explains this?
Cell A is a sphere with a radius of 2 µm, and cell B is a sphere with a radius of 6 µm. How does the surface area-to-volume ratio of cell A compare with that of cell B? (Surface area of a sphere = 4πr²; volume of a sphere = (4/3)πr³)
0 of 4 answered