1Why volume outruns surface

Surface area grows with the square of length; volume grows with the cube of length. Double the side of a cube and its surface area becomes 4 times larger, but its volume becomes 8 times larger. So its SA:V halves.

Figure 1 shows four cubes with sides of 1, 2, 4 and 8 units. The ratio goes 6 : 1, 3 : 1, 1.5 : 1, 0.75 : 1. Each doubling of the side halves the ratio. Read the curve: it falls steeply at small sizes and flattens out, which is why the drop matters most when a small cell starts to grow.

Bigger means a smaller surface area-to-volume ratio: less membrane per unit of cytoplasm to serve.

2Doing the calculation

The formula sheet gives you the shapes. Choose the one closest to the cell, calculate surface area and volume separately, then divide.

Worked example 1: a cell doubles its radius. A spherical cell has a radius of 10 µm.

$$SA = 4\pi(10)^2 = 1{,}257\ \mu\text{m}^2 \qquad V = \tfrac{4}{3}\pi(10)^3 = 4{,}189\ \mu\text{m}^3$$ $$\frac{SA}{V} = \frac{1{,}257}{4{,}189} = 0.30\ \mu\text{m}^{-1}$$

It grows to a radius of 20 µm: $SA = 5{,}027\ \mu\text{m}^2$ and $V = 33{,}510\ \mu\text{m}^3$, so $SA/V = 0.15\ \mu\text{m}^{-1}$. The surface area went up 4 times, the volume 8 times, and the ratio halved. (For a sphere the ratio simplifies to $3/r$, a quick check.)

Worked example 2: shape, not just size. Two cells have the same volume, 512 µm³. One is a cube with sides of 8 µm: $SA = 6 \times 8^2 = 384\ \mu\text{m}^2$, so $SA/V = 0.75\ \mu\text{m}^{-1}$. The other is long and thin, a rectangular solid 32 µm by 4 µm by 4 µm:

$$SA = 2(32)(4) + 2(32)(4) + 2(4)(4) = 544\ \mu\text{m}^2 \qquad \frac{SA}{V} = \frac{544}{512} = 1.06\ \mu\text{m}^{-1}$$

Same volume, 42% more surface. Flattening or stretching a cell raises its ratio without making it smaller.

The ratio has units of length⁻¹ (µm⁻¹ here). Many questions write it as a ratio instead, such as 0.75 : 1. Both are accepted; just be consistent when comparing.

3What a falling ratio does to a cell

As a cell gets larger:

• its volume rises fastest, so the demand for oxygen and nutrients, and the production of waste, rise fastest;
• its surface area, the only route in and out, rises more slowly;
• so the supply per unit of volume falls, and distances from the membrane to the centre of the cell get longer, so diffusion to the middle takes longer.

That is why most cells are microscopic. Smaller cells have a higher SA:V and exchange materials with their surroundings more efficiently than larger ones.

A model of this is a cube of agar jelly with a coloured indicator, dropped into a solution that diffuses in and changes the colour. In an invented run, the solution reaches 0.3 cm in from every face in 10 minutes:

The diffusion distance is the same for every cube, but in a big cube that distance reaches a smaller fraction of the volume. The centre of a large "cell" would be short of supplies.

4Getting more surface without getting smaller

Cells that must exchange a lot of material raise their ratio with more complex structures.

• Membrane folds. Cells lining the small intestine (gut epithelial cells) carry thousands of tiny finger-like folds of plasma membrane, microvilli, on the side facing the gut. They add a great deal of surface for absorbing digested food while adding very little volume.
• Long, thin projections. A root hair is a long tube growing out of a single root cell into the soil, which multiplies the area for taking up water and mineral ions.
• Folds inside the cell. The folded inner membrane of a mitochondrion (2.1) is the same idea inside an organelle.

The same logic works at the level of structures: plant leaves are thin and flat, and gas moves through pores called stomata, opened and closed by guard cells, giving a large surface for gas exchange.

5Whole organisms: heat and metabolic rate

The rule scales up. As an organism gets larger, its SA:V decreases. Heat is lost through the surface, so a small mass exchanges proportionally more heat with its surroundings than a large mass. As mass increases, both the SA:V and the rate of heat exchange per unit mass decrease.

That links to metabolism. A small mammal loses heat fast for its size, so it must release energy fast to stay warm: its metabolic rate per unit body mass is high. A large mammal loses heat slowly for its size, and its metabolic rate per kilogram is lower. In general, the smaller the organism, the higher its metabolic rate per unit body mass.

An invented data set shows the pattern:

Each kilogram of species W uses about 13 times as much oxygen per minute as each kilogram of species Z. Note what the data do not say: the large animal uses far more oxygen in total; it is the rate per kilogram that is lower.