Cell size and the surface area to volume ratio
Everything a cell needs (oxygen, food, water) enters through its surface, the cell membrane. Everything it makes (waste, heat, products) leaves the same way. But the volume decides how much it needs.
For a cube of side s: surface area = 6s², volume = s³, so SA:V = 6/s. Double the side: surface ×4, volume ×8, ratio halves.
| Side (µm) | SA (µm²) | V (µm³) | SA:V |
|---|---|---|---|
| 1 | 6 | 1 | 6 |
| 2 | 24 | 8 | 3 |
| 3 | 54 | 27 | 2 |
| 6 | 216 | 216 | 1 |
For a sphere SA = 4πr², V = ⁴⁄₃πr³, so SA:V = 3/r. Same lesson: bigger means a smaller ratio.
A low ratio means slow exchange and long diffusion distances to the centre. So cells stay small and divide when they grow. Cells that must be large change shape instead: long and thin (nerve cells), flat (red blood cells, a biconcave disc), or folded (microvilli in the gut). Big organisms use many small cells plus exchange organs (lungs, gills, villi) and transport systems (blood).
Compartments: why eukaryotic cells have rooms
Prokaryotes (bacteria and archaea) are small (about 1–5 µm). They have DNA in a region called the nucleoid, ribosomes and a cell membrane, but no membrane-bound organelles. Some fold their cell membrane inward to gain area.
Eukaryotes (plants, animals, fungi, protists) are usually 10–100 µm. Inside, membranes make compartments: nucleus, endoplasmic reticulum (ER), Golgi, lysosomes, vacuoles, mitochondria and chloroplasts.
- Compartments separate reactions that would clash, e.g. digestive enzymes stay inside lysosomes.
- Each room keeps its own conditions (pH, enzymes, concentrations).
- Internal membranes add a huge surface area for reactions and for enzymes to sit on.
The inner membranes (ER, nuclear envelope) are thought to have formed when the cell membrane folded inward. Mitochondria and chloroplasts have a different story: endosymbiosis.
Endosymbiotic theory
About 1.5–2 billion years ago, a larger host cell engulfed an aerobic bacterium (one that uses oxygen) but did not digest it. The two helped each other: the bacterium got food and shelter, the host got lots of ATP. Over time the bacterium became the mitochondrion. Later, some cells engulfed a photosynthetic bacterium (like a cyanobacterium), which became the chloroplast.
Evidence:
- Mitochondria and chloroplasts have two membranes (the inner one from the bacterium, the outer from the engulfing).
- They have their own circular DNA, like bacteria.
- They have their own 70S ribosomes, like bacteria (the cytoplasm has 80S).
- They divide by binary fission, on their own, like bacteria.
- Their DNA sequences are closest to those of certain living bacteria.
Cellular energy: mitochondria and chloroplasts
Cells run on ATP, a small molecule that carries usable energy.
- Chloroplasts (plants, algae) capture light energy on the thylakoid membranes (stacked into grana) and use it in the stroma to build glucose from CO₂ and water: photosynthesis.
- Mitochondria (almost all eukaryotes) break down glucose with oxygen to make lots of ATP: aerobic respiration. The inner membrane is folded into cristae, surrounded by the fluid matrix.
Both organelles make ATP on membranes, so more membrane = more ATP. Folding gives a large area in a small space. Busy cells, like heart muscle cells, have many mitochondria with many cristae. Energy flows: light → glucose (chloroplast) → ATP (mitochondrion) → work (movement, transport, building).
Try it: the agar or potato cube test
Cut potato (or beetroot) cubes of side 1 cm, 2 cm and 3 cm. Put them in salty water or in diluted food colour for 15 minutes. Cut each cube in half. The colour reaches the centre of the small cube, but not of the big one. Work out SA:V for each (6, 3, 2 per cm) and match it to how far the colour got. Then check your numbers with the slider in the 3D.
Key formulas and definitions
- Cube: SA = 6s², V = s³, SA:V = 6/s
- Sphere: SA = 4πr², V = ⁴⁄₃πr³, SA:V = 3/r
- Double the size: surface × 4, volume × 8, SA:V ÷ 2
- Mitochondrion: glucose + O₂ → CO₂ + H₂O + ATP
- Chloroplast: CO₂ + H₂O + light → glucose + O₂
- Endosymbiosis evidence: 2 membranes, circular DNA, 70S ribosomes, binary fission
Worked examples
1. Find SA, V and SA:V for a cube-shaped cell of side 4 µm.
SA = 6 × 4² = 96 µm². V = 4³ = 64 µm³. SA:V = 96/64 = 1.5 (or 6/4 = 1.5).
2. A cube of side 3 µm is cut into 27 cubes of side 1 µm. Compare the total surface area.
Before: 6 × 3² = 54 µm². After: 27 × 6 × 1² = 162 µm². Volume stays 27 µm³. Surface triples, so SA:V goes from 2 to 6.
3. A spherical cell has radius 5 µm. Find SA:V.
SA:V = 3/r = 3/5 = 0.6 µm⁻¹. (SA = 4π × 25 ≈ 314 µm², V = ⁴⁄₃π × 125 ≈ 524 µm³, 314/524 ≈ 0.6.)
4. Why does a red blood cell have a flat, dented (biconcave) shape?
The flat shape gives a much larger surface for its volume than a sphere, and no point is far from the surface, so oxygen diffuses in and out quickly.
5. Give three pieces of evidence that chloroplasts were once free-living bacteria.
They have a double membrane, their own circular DNA and 70S ribosomes like bacteria, and they divide by binary fission inside the cell.
Common mistakes
- Saying a bigger cell has a bigger SA:V. The surface is bigger, but the ratio is smaller, because volume grows faster.
- Forgetting the units: SA:V for a cube of side s µm is 6/s per µm (µm⁻¹).
- Thinking prokaryotes have no ribosomes or no DNA. They have both; they just lack membrane-bound organelles like a nucleus.
- Saying mitochondria 'make energy'. Energy cannot be created; mitochondria convert the chemical energy in glucose into ATP.