The Isoperimetric Problem — The Shape of Maximum Area

Of all closed curves with a given perimeter, which one encloses the maximum area? The circle. This is the isoperimetric inequality, suspected since antiquity and proved rigorously only in the 19th century. Queen Dido, founding Carthage, was given as much land as she could enclose with a hide — she cut it into thin strips, formed a semicircle against the sea, and enclosed the most possible land. Nature also solves isoperimetric problems: soap bubbles are spheres because a sphere minimises surface area for a given volume.