3.3.1 - Surface area to volume ratio

Why volume outgrows surface area

Scale a body up to twice the length and its surface area grows fourfold, but its volume grows eightfold.

Materials cross an organism's surface, but they are used or produced by the cells that fill its volume. So the useful question is not how much surface there is, but how much surface each unit of volume gets.

As an object gets bigger and keeps a similar shape, its volume increases faster than its surface area, so SA:V decreases. Surface area increases with the square of the linear dimensions, while volume increases with the cube.

This applies to a cell, an organ or a whole organism. Scaling up a cube shows the effect:

The graph plots the last column against edge length. Notice that the curve falls steeply at small sizes, then flattens.

[DIAGRAM: asset_name: 3.1 - Surface area to volume ratio - Diagram 1; asset_slug: 3.1 - Surface area to volume ratio - Diagram 1; recommended_method: retained_png; caption: Surface area divided by volume falls as cube edge length increases, all lengths in centimetres.; description: Surface area divided by volume falls as cube edge length increases; all lengths in centimetres.]

When the edge doubles from 1 cm to 2 cm, surface area is multiplied by 4 but volume by 8, so the SA:V ratio halves from 6:1 to 3:1. By the time the edge reaches 6 cm, the ratio has fallen to 1:1.

So as size increases, each unit of volume has less surface supplying it, and exchange across that surface is less able to meet the needs of the cells inside. To compare two sizes, calculate both ratios, then say what the change means for surface per unit volume.

Calculating surface area to volume ratios

You can find SA:V for any regular cell shape, once you treat the cell as a simple geometric model.

State the model: real cells are rarely perfect spheres or cylinders, and only surfaces exposed to the exchange medium contribute directly to exchange. In the formulas below, $r$ is the radius, half the diameter. They build on the circle, which has circumference $C=2\pi r$ and area $A=\pi r^2$.

Because area and volume have different units, the value depends on the length unit: the same cell has $S/V=600$ mm⁻¹. So compare ratios only after using the same length unit.

Cylinders: remember the ends

Unroll the curved surface of a closed cylinder and you get a rectangle: its width is the circumference $2\pi r$ and its height is $h$. Then add the two circular ends. For a model cell with radius 2 μm and length 10 μm:

• Circumference and end area: 12.56 μm and 12.56 μm²
• Curved area: 12.56 × 10 = 125.6 μm²
• Total surface area, adding both ends: 150.72 μm²
• Volume: 125.6 μm³, so $S/V=1.20$ μm⁻¹

Exclude the ends only if the question states that they do not exchange material.

Rectangular prisms: why shape matters

A block 4 × 2 × 1 mm has $S=2(8+4+2)=28$ mm² and $V=8$ mm³, so $S/V=3.5$ mm⁻¹. A cube of the same volume has edge 2 mm and $S/V=24/8=3.0$ mm⁻¹.

So the flatter shape has an exchange advantage over a cube of equal volume: more exchange area per unit volume, and a shorter path to central cells. Keep extra digits during a calculation and round only the final answer, for example to two significant figures.

Exchange in small and large organisms

An amoeba has no lungs, gills or blood: diffusion across its surface supplies its oxygen.

Small organisms such as Amoeba, other single-celled protists and bacteria have a SA:V ratio large enough for exchange by simple diffusion across the body surface. Every part of the cell is close to the cell-surface membrane, so the diffusion pathway is extremely short, and they need no specialised exchange surfaces or transport systems.

Larger, multicellular organisms have a lower SA:V ratio, which causes two linked problems:

• Insufficient surface area: the outer body surface cannot absorb or release materials fast enough to serve the much greater volume of metabolically active tissue.
• Increased diffusion distance: even if enough material crossed the surface, diffusion alone would take far too long to reach cells deep in the body.

Size is not the only factor: an elongated or flattened shape can keep the diffusion path short even in a relatively large organism. And large organisms still use diffusion at their exchange surfaces. What falls short is diffusion across the outer body surface alone.

Cover the table, then explain the difference from memory.