Question P.419: Among all the closed curves of same length, the circle is th......

We consider a circle with circumference C, and a closed curve with the same length. In the circle, we inscribe a regular polygon with n sides, and let S_{n},\ P_{n} denote its area and perimeter respectively. In the closed curve we inscribe another polygon with n sides, and let s_{n},\ p_{n} denote its area and perimeter. The result of exercise 418b tells us that

(1)            {\frac{S_{n}}{P_{n}^{2}}}\geq{\frac{s_{n}}{p_{n}^{2}}}

for any n.

Now let n increase without bound. Then we can choose the vertices of the polygon inscribed in the closed curve in such a way that the length of each of the sides approaches 0. Thus \mathbf{}P_{n} approaches C, and \mathbf{}p_{n} approaches c. Also, Sn approaches S and \mathbf{}s_{n} approaches s, where S and s denote the area bounded by the circle and the other closed curve, respectively. From (1), it follows that {\frac{{S}}{C^{2}}}\geq{\frac{s}{c^{2}}}, so that S ≥ s.

Thus, of all closed curves with a given length, none can enclose a larger area than that of the circle with the given length as circumference.

Note. It remains to prove that no closed curve other than the circle can also enclose the largest possible area, of all curves with a given length. Indeed, it is possible, in light of the inequality S ≥ s that there is some other closed curve, not a circle, with length C and area S (that is, s = S for this curve). In fact, a more advanced investigation would show that there is no such curve.