Is the area of a convex polygon equal to the area of a circle with the same perimeter of the polygon? I guess that it's possible, take for example an square, I guess that it's borders could be deformed to form a circle and that their areas would be the same but such condition holds only for convex polygons. I guess I've read some theorem about it in the past but I don't remember it now.
The area of a circle is greater than the area of any polygon with the same perimeter. Equivalently, the circumference of a circle is smaller than the perimeter of any polygon with the same area. For details and references, please see the Wikipedia article on the Isoperimetric Inequality.
Well, why don't you just try the circle and square first?
Take a circle with radius 1, which has an area of $\pi$ and perimeter of $2\pi$. A square with the same perimeter would have a side length of $\pi/2$, hence an area of $\pi^2/4$.
Since $\pi$ isn't equal to $\pi^2/4$, you already have a counterexample.