Surface area to volume ratio

In a nutshell

Every substance an organism needs (oxygen, glucose) enters across its surface, and every waste product leaves the same way, mostly by diffusion.

How much surface an organism has relative to its volume decides whether simple diffusion across the body surface is enough, or whether it needs a specialised exchange surface and a transport system.

This subtopic is about the surface area to volume ratio: how it changes with size, and why that change forces larger organisms to adapt.

Assumed knowledge: Diffusion.

Core content

What the ratio is, and why it matters

The surface area to volume ratio (SA:V) is the total surface area of an organism or structure divided by its total volume.

• The surface area sets how fast substances can be exchanged with the environment (the area available for diffusion).
• The volume sets how much substance the organism needs to supply, and how much waste and heat it produces.

So SA:V compares supply capacity against demand. A high ratio means plenty of surface for the volume it serves; a low ratio means the surface struggles to service the volume behind it.

How the ratio changes with size

As an organism gets larger, its volume increases faster than its surface area, so as size increases, the surface area to volume ratio decreases.

The clearest way to see this is with cubes, where surface area depends on the side length squared but volume on the side length cubed:

Plotted, the ratio falls away steeply as the object gets bigger:

The reason is the different rates of increase: double the side length and the surface area goes up 4-fold, but the volume goes up 8-fold. Volume always wins.

Still don't get it? · why the ratio falls as size increases

Think about wrapping presents. One big box takes relatively little wrapping paper for the amount of stuff inside it. Wrap that same amount of stuff as lots of tiny boxes instead, and you use far more paper for the same contents. Small things have a lot of "outside" for their "inside"; big things have very little.

Now build it back up in numbers. The "outside" is the surface, and it grows with length × length (squared). The "inside" is the volume, and it grows with length × length × length (cubed). Because you multiply by the length one extra time, the inside grows faster than the outside every time you scale up.

So when you make an organism bigger, the volume it has to supply races ahead of the surface it has to supply it through. In exam wording: as size increases, the surface area to volume ratio decreases, so a large organism has a smaller SA:V than a small one.

Why a small SA:V is a problem for large organisms

In a small organism, every cell is close to the surface, so the diffusion distance is short and diffusion across the body surface alone supplies the whole organism fast enough.

In a large organism, cells deep inside are a long way from the surface, and the surface is small relative to the volume it must serve. Diffusion across the outer surface is now too slow over too long a distance to meet demand.

Larger organisms overcome this in three ways, all of which raise the effective surface area or shorten the distance:

• Changes to body shape: becoming flat or thin (e.g. a flatworm) keeps every cell close to the surface, so the SA:V stays high enough for diffusion.
• Specialised exchange surfaces: organs such as lungs, gills and the small intestine provide a huge surface area folded into a small volume.
• Mass transport systems: a circulatory system carries substances between the exchange surface and the body's cells, maintaining steep diffusion gradients.

Adaptations of exchange surfaces

Where an organism needs to increase exchange, its surfaces are adapted to raise the surface area to volume ratio and shorten the diffusion pathway. Where it needs to conserve heat, the opposite shape helps.

Always tie the ratio to the feature and the function: a folded surface gives a larger surface area to volume ratio, which gives a faster rate of diffusion for exchange.

The link to metabolic rate

Metabolic rate is the rate at which an organism uses energy, and it is usually measured as the rate of oxygen consumption (oxygen is used in respiration, which releases the energy).

A small mammal has a large surface area to volume ratio, so it loses heat rapidly across its surface. To keep its body temperature constant, it must respire faster to release heat, giving it a high metabolic rate and therefore a high demand for oxygen and food per gram of body mass.

A large mammal has a small SA:V, loses heat slowly, and has a lower metabolic rate per unit mass.

Still don't get it? · linking SA:V to metabolic rate

Picture a small cup of tea and a big pot of tea, both poured boiling. The small cup goes cold much faster, because it has a lot of surface for its small amount of liquid. That is exactly a small animal: lots of surface for its little volume, so heat escapes fast.

Now the animal is warm-blooded, so it cannot just let its temperature drop. To stay warm it has to keep making heat, and it makes heat by respiring (breaking down glucose, using oxygen). The faster it loses heat, the faster it has to respire to replace it.

"How fast an organism respires / uses energy" is its metabolic rate, and because respiration uses oxygen, we measure metabolic rate as the rate of oxygen consumption. So the exam chain is: small size, large SA:V, rapid heat loss, high metabolic rate, high oxygen consumption per gram.

Required practical skill: modelling diffusion with agar cubes

You can model the effect of SA:V on diffusion using cubes of agar containing an indicator (for example, agar dyed with an alkali indicator, placed in dilute acid).

• Cut agar cubes of different side lengths but the same shape, so only SA:V differs.
• Place them in the acid at the same time and leave for a fixed time.
• Cut each cube open and measure how far the colour change (the acid) has diffused in, or record the time for the colour to change all the way to the centre.

Result and interpretation:

• The smallest cube has the largest SA:V and the shortest distance from surface to centre, so the acid reaches its centre first and it changes colour throughout soonest.
• The largest cube has the smallest SA:V, so in the same time the acid has changed only the outer layer.

This models why large organisms cannot rely on diffusion across the surface alone: as SA:V falls, the same fixed rate of surface exchange supplies proportionally less of the interior.

Worked examples

Calculation: compare the SA:V of two cube-shaped blocks.

A block of side 2 cm and a block of side 4 cm. Surface area of a cube is $6s^2$ and volume is $s^3$.

Block of side 2 cm: $SA = 6 \times 2^2 = 24\ \text{cm}^2 \qquad V = 2^3 = 8\ \text{cm}^3$ $SA:V = \frac{24}{8} = 3 \quad \Rightarrow \quad 3:1$

Block of side 4 cm: $SA = 6 \times 4^2 = 96\ \text{cm}^2 \qquad V = 4^3 = 64\ \text{cm}^3$ $SA:V = \frac{96}{64} = 1.5 \quad \Rightarrow \quad 1.5:1$

Doubling the side length halved the ratio (3:1 down to 1.5:1). Always write the ratio the correct way round, surface area to volume, and state it as a number "to 1".

Model 3-mark answer: "Explain why a mouse has a higher rate of oxygen consumption per gram than an elephant."

1. A mouse is smaller, so it has a larger surface area to volume ratio than an elephant.
2. So it loses heat faster across its body surface (relative to its volume).
3. To maintain a constant body temperature it must respire faster to release heat, so it has a higher metabolic rate and uses more oxygen per gram.

Notice this is a three-step causal chain, and the first point must be the ratio, not "a bigger surface area".

Common exam mistakes

• Writing "surface area" when the mark needs "surface area to volume ratio". A flatworm exchanges gases efficiently because of its large SA:V, not just a large surface area, and answers that drop the "to volume ratio" lose the mark.
• Giving the ratio the wrong way round (volume to surface area), or quoting a ratio "in reverse". It is surface area : volume, so a small cell has a large number (e.g. 6:1), not a small one.
• Being asked to describe the relationship and instead giving one example ("a large organism has a small ratio"). State it as a relationship: "as size increases, the surface area to volume ratio decreases".
• Comparing absolute surface areas ("a mouse has a smaller surface area than an elephant"). A larger organism has a larger total surface area but a smaller SA:V; only the ratio is the point.
• Stating a large SA:V but not linking it to a body feature and its function. The marks come from the chain: feature (e.g. folded surface), larger SA:V, faster diffusion for exchange.
• On metabolic rate, saying oxygen "produces energy" or confusing breathing with respiration. Oxygen is used in respiration, which releases energy; energy is not made.
• Confusing the amount of heat lost with the rate relative to size. A small mammal does not lose more heat in total; it loses heat rapidly for its volume because of its large SA:V, which is why it needs a high metabolic rate.

Key definitions

• Surface area to volume ratio: the total surface area of an organism or structure divided by its total volume.
• The size relationship: as the size of an organism increases, its surface area to volume ratio decreases (smaller organisms have a larger ratio).
• Metabolic rate: the rate at which an organism uses (or releases) energy, usually measured as the rate of oxygen consumption.

Specification

• I can state the relationship between the size of an organism or structure and its surface area to volume ratio.
• I can calculate the surface area to volume ratio of cells or structures of different shapes from their dimensions.
• I can explain how changes to body shape and the development of exchange and transport systems in larger organisms are adaptations to a decreasing surface area to volume ratio.
• I can explain the relationship between surface area to volume ratio and metabolic rate.

Related notes

• Transport across cell membranes
• Gas exchange
• Digestion and absorption
• Mass transport