Assume we want to maximize the area of such of a polygon. It's easy to show that from a particular $n$-gon we can obtain another one with larger area and which is convex. Now assume we have another convex $n$-gon with the same sides as our initial one, but larger area. On the sides construct the arcs of the circle in which the first $n$-gon was inscribed (this is where I need convexity: I don't want the arcs to overlap). We get a figure with the same perimeter as that of the circle, but larger area, but it's well-known that among all shapes of the same perimeter, the circle is the one with the largest area, so we have a contradiction.

I just realized that we don't need the convexity: if the second polygon is not convex, then the perimeter and area of the figure we get after drawing the circular arcs on its sides are even smaller, and that's Ok from our point of view (we're trying to show that the area can't be larger than that of the circle with a perimeter $\ge$ that of the figure). Is this Ok?