The question is in the title: Is there any closed curve whose area is proportional to its perimeter?
If not, why is it so? Can it be proved?
I tried all the simple shapes I know, but couldn't find a solution.
Is there any closed curve whose area is proportional to its perimeter
curvesgeometry
Related Solutions
Counter-example without words (except for these):
By the Jordan curve theorem, a simple closed curve separates the plane into two path-connected regions: a bounded region (the points interior to the curve) and an unbounded region (the points exterior to the curve). You are asking if the bounded region can have infinite area. The answer is no since the curve is a compact and hence bounded subset of the plane. So you can find a rectangle that includes the curve and hence includes its interior, implying that the area of the interior is at at most equal to the (finite) area of the rectangle.
Best Answer
You cannot get more proportional than equality. There is a family of rectangles whose perimeters are equal to their areas:
They have sides for the form $x + \sqrt{x^2-4x}$ and $x - \sqrt{x^2-4x}$ with $x \ge 4$
and so perimeters and areas each equal to $4x$.
Nice examples from this family are