What is the surface area to volume ratio?
Surface area (SA) is the total area of the outside. Volume (V) is the space inside. The SA : V ratio is SA divided by V. It tells you how much outside there is for each cubic unit of inside.
For a cube with side L: SA = 6 × L², V = L³, so SA : V = 6L² ÷ L³ = 6 ÷ L. Write it as a number “to 1”, e.g. 3 : 1.
- Side 1 cm: 6 ÷ 1 = 6 : 1
- Side 2 cm: 24 ÷ 8 = 3 : 1
- Side 3 cm: 54 ÷ 27 = 2 : 1
Rule: as an object gets bigger, its SA : V ratio gets smaller, because area grows as length² but volume grows as length³. Other shapes follow the same rule: for a sphere SA : V = 3 ÷ r; for a cylinder, include the two ends and the curved side (2πr² + 2πrh).
Why size matters for exchange
Every living thing must take in oxygen and nutrients and get rid of carbon dioxide, urea and heat. These pass through the surface, but they are used or made by the whole volume.
- Small organisms (bacteria, amoeba): large SA : V and a short diffusion distance to every part. Simple diffusion across the body surface is fast enough.
- Large organisms: small SA : V and long distances to inner cells. Diffusion alone would be far too slow, and the surface would be too small to supply all the volume.
Fick's idea in words: diffusion is faster with a bigger area, a steeper concentration difference and a shorter distance.
How large organisms solve the problem
- Shape: flat or thin bodies (flatworms, leaves) keep cells close to the surface.
- Specialised exchange surfaces with folds and thin walls: alveoli in lungs, villi and microvilli in the small intestine, gill filaments and lamellae in fish, tracheoles in insects, root hairs and spongy mesophyll in plants.
- Mass transport systems: blood and a heart in animals; xylem and phloem in plants. They keep a steep concentration gradient by carrying substances away.
- Ventilation: breathing or water flow over gills brings fresh supplies.
A good exchange surface is: large in area, thin (short distance), kept moist, with a good blood supply and ventilation.
Size, heat loss and metabolic rate
Mammals and birds keep a constant body temperature. Heat is lost through the surface. A small animal has a large SA : V, so it loses heat fast for its size. To replace it, it needs a high metabolic rate per gram: more respiration, more food and more oxygen per gram of body. A shrew eats about its own body mass in food each day; its heart can beat over 800 times a minute.
Large animals have a small SA : V, keep heat well and have a lower metabolic rate per gram, but can overheat. African elephants flap large thin ears to lose heat. Animals in cold places tend to be large and compact with small ears; those in hot deserts tend to have long ears and thin bodies. A higher metabolic rate also means a greater need for efficient gas exchange.
Try it: the agar cube practical
At home: cut a potato or a jelly block into cubes of 1, 2 and 3 cm. Put them in coloured water or vinegar with a pinch of food colour for 10 minutes, then cut each in half. The small cube is coloured right through; the big one still has a pale centre. In a lab, pink agar with an indicator turns colourless as acid diffuses in; you can time how long each cube takes. In the 3D, slide the cube side and predict the ratio first.
Key formulas and definitions
- Cube: SA = 6L², V = L³, SA : V = 6 ÷ L
- Cuboid: SA = 2(lw + lh + wh), V = l × w × h
- Sphere: SA = 4πr², V = (4/3)πr³, SA : V = 3 ÷ r
- Cylinder: SA = 2πr² + 2πrh, V = πr²h
- SA : V written as x : 1 (divide SA by V)
- Rate of diffusion ∝ (surface area × concentration difference) ÷ diffusion distance
Worked examples
1. Find the SA : V ratio of a cube with 4 cm sides.
SA = 6 × 4² = 96 cm². V = 4³ = 64 cm³. SA : V = 96 ÷ 64 = 1.5 : 1.
2. A cuboid cell is 2 µm × 2 µm × 5 µm. Find its SA : V.
SA = 2(2×2 + 2×5 + 2×5) = 2(4 + 10 + 10) = 48 µm². V = 2 × 2 × 5 = 20 µm³. SA : V = 48 ÷ 20 = 2.4 : 1.
3. A spherical cell has radius 3 µm. Find its SA : V (π = 3.14).
SA = 4πr² = 4 × 3.14 × 9 = 113.04 µm². V = (4/3)πr³ = (4/3) × 3.14 × 27 = 113.04 µm³. SA : V = 1 : 1 (check: 3 ÷ r = 3 ÷ 3 = 1).
4. A 2 cm cube is cut into eight 1 cm cubes. What happens to the total surface area and the SA : V?
Before: SA = 24 cm², V = 8 cm³, ratio 3 : 1. After: 8 cubes × 6 cm² = 48 cm², V still 8 cm³, ratio 6 : 1. The surface doubles and the ratio doubles. This is why chewing food speeds up digestion.
5. Cube A has side 1 mm, cube B has side 10 mm. How many times bigger is B's ratio than A's, or smaller?
A: 6 ÷ 1 = 6. B: 6 ÷ 10 = 0.6. B's ratio is 10 times smaller, while its volume is 1000 times larger.
6. A shrew (8 g) and a horse (500 kg). Which has the higher oxygen use per gram, and why?
The shrew. It is much smaller, so its SA : V is much larger and it loses heat faster per gram. To keep its body temperature it must respire faster per gram, so it uses more oxygen per gram.
7. A flatworm is 0.5 mm thick and 10 mm long. Why does it not need lungs or blood?
Its flat shape gives a large SA : V and no cell is more than about 0.25 mm from the surface, so oxygen can reach every cell by diffusion alone.
Common mistakes
- Thinking bigger animals have a bigger SA : V. They have more surface in total, but less surface per unit volume.
- Forgetting units: SA is in units², V in units³, and the ratio has no unit (write x : 1).
- Counting only 1 or 4 faces of a cube. A cube has 6 faces; a cuboid has 3 pairs of different faces.
- Saying large organisms “cannot do diffusion”. They still use diffusion, but only across short distances at special exchange surfaces.