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The Isoperimetric Inequality states that the area enclosed by any simple, closed, planar curve having perimeter P is less than or equal to , where equality holds if and only if the curve is a circle. More informally, this means that of all the curves having the given perimeter P, the circle has the greatest enclosed area.
The Isoperimetric Inequality has a rich and interesting history. A version of this problem in which the curves were restricted to the class of rectangles was solved in Euclid's Elements. Pappus of Alexandria, who lived about 300 A.D. proved that the area of a circle is greater than that of any polygon of the given perimeter. The famous 19th century geometer Jacob Steiner claimed to have a geometric proof of the inequality. However, Karl Wierstrass showed that the proof was not completely rigorous. It was Wierstrass himself, who published the first rigorous proof, using analytic methods, in 1870.