Problem statement: $a)$ Show that the supreme of the areas of the convex regular polygons with prefixed perimeter p, is given by the area of the corresponding circle with perimeter p. $b)$ Show that between such polygons the greater the number of sides, the greater the area.
I can't solve this problem. I have tested the following:
$a)$ Using the formula for the area of a regular polygon and the Pythagorean theorem, I have arrived at that the area is
$$A(\theta(N)) = \frac{p \cdot l}{4 \tan(\theta(N))} $$
where $\theta(N)=\frac{360}{2N}$ , p is the perimeter and l de lenght of a side. I think the best way would be to take limits when N tends to infinity but I would not know how to do it.
$b)$ As the function $ A (\theta) $ is differentiable, I think it would be best to study monotony and see that it is increasing. But I wouldn't know how to do it.