Diffusion: Students should be able to calculate and compare surface area to volume ratios.

The idea

To calculate a surface area to volume ratio, work out the total surface area and the volume of the object using the same unit of length, then divide the surface area by the volume and write the result in the form surface area : volume, often simplified so the volume is 1. For a cube of side 2 cm, each face has area 2 cm × 2 cm = 4 cm², so six faces give 24 cm²; the volume is 2 cm × 2 cm × 2 cm = 8 cm³; the ratio is 24 : 8, which simplifies to 3 : 1. For a cube of side 4 cm, the surface area is 96 cm² and the volume is 64 cm³, giving 96 : 64, which simplifies to 1.5 : 1. The smaller cube has the larger ratio, showing that as size increases the ratio decreases.

What earns the marks

• Calculate total surface area correctly for the shape given, for example 6 × side² for a cube.
• Calculate volume correctly for the shape given, for example side³ for a cube.
• Divide surface area by volume, or write the two values as a ratio, keeping the same length unit throughout.
• Simplify the ratio to its simplest form, for example 24 : 8 becomes 3 : 1.
• Compare two or more ratios and state which object has the larger surface area to volume ratio and what that means for diffusion.

Common mistakes and how to put them right

• Mixing units, such as using centimetres for length but millimetres for volume.

  Put it right: Convert all lengths to the same unit before calculating.

• Forgetting to multiply by the number of faces when finding the surface area of a cube, giving 4 cm² instead of 24 cm² for a 2 cm cube.

Exam tips

• Show each step: surface area, volume, then the ratio, so method marks can be awarded even if the final simplification slips.
• Simplify ratios by dividing both sides by the smaller value, and state the ratio clearly, for example 3 : 1.
• When comparing, quote both calculated ratios and then say which is larger and why that matters for diffusion.