In ancient times the extent of a city or an armed camp was often given in terms of its perimeter (so that a town would be described as requiring so many thousand paces to walk round). In the same way, according to Proclus, certain socialistic communities used to divide land so that each family received a plot of equal perimeter and it may have been in this context that it was first realised that a square contains a much greater area than a long thin rectangle of the same perimeter.

Once it was understood that figures with the same perimeter may contain different areas it was natural to ask whether there exists a figure of maximum area. It is not hard to guess that the answer is a circle but a guess is not a proof. The isoperimetric problem thus asks for a proof that among all figures of equal perimeter the circle has greatest area.

This question formed the subject of one of the last substantial investigations of the golden age of Greek geometry. In it Zenodorus proved that the circle has greater area than any polygon of the same perimeter.

We might expect that a purely geometrical approach could not go much further in the absence of precise notions of area and length. However, in 1841 Steiner showed how simple geometric considerations could be used to prove the following theorem.