A theory of the elastic behaviour of thin rubber films is developed to describe inflation of pressure-limited tracheal tube cuffs (a) outside and (b) inside rigid cylindrical model tracheas. At each stage, assumptions and results were successfully checked against experiment. The theory predicts cylinder wall versus intracuff pressure. This allows, for any suitable cuff (of which there are many), the determination of a single intracuff pressure that ensures a satisfactory seal, for any diameter within the human adult range, the wall pressure staying below some chosen safety limit.