Question P.418b: Among all polygons with the same number of sides and the sam......
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.
Suppose ABCDE… is the polygon referred to, with a given number of sides and a given perimeter, and with the largest possible area. (We assume that such a polygon exists.) We have solved this problem completely for triangles in exercise 416, so we can assume that our polygon has at least four sides.
Take four consecutive vertices A, B, C, D of our polygon. If these four vertices did not lie on the same circle, then we could increase the area of our polygon by replacing quadrilateral ABCD with cyclic quadrilateral A B^{\prime}C^{\prime}D (keeping vertices A and D fixed), by the last result of exercise 417. We can repeat this argument for any four consecutive vertices of the polygon, so the polygon of largest area must be cyclic.
Now suppose that two adjacent sides of the polygon, say AB and BC, are not equal. Then we can replace triangle ABC with an isosceles triangle with the same perimeter, but having a larger area (by the result of exercise 416).
Since the quadrilateral of largest area is cyclic, and has equal sides, it must be regular.
We can phrase this result in terms of the ratio S : p^{2}. If we compare a regular polygon with a non-regular polygon with the same number of sides and the same perimeter, this ratio will be larger for the regular polygon, since its area will be larger. And since all regular polygons with the same number of sides are similar, this ratio will not depend on the size of the regular polygon.
Note. In more advanced work, we can obtain this result even without assuming that the polgyon of largest area exists.