Among all polygons with the same number of sides and the same perimeter, the largest is the regular polygon. (Assuming that a polygon of maximum area exists, we can use the preceding exercises and Exercise 331 to show that this polygon must be regular.) The result can be restated as follows: If S is the area of a polygon, and p its perimeter, the ratio \frac{S}{p^{2}} is larger for a regular polygon than for an irregular polygon with the same number of sides.