# Is the area of a convex polygon equal to the area of a circle with the same perimeter of the polygon?

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.

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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.
A circle with radius $1$ has a perimeter of $2\pi$. –  JiK Aug 10 '14 at 20:38