An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible? Problem: An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible?
To answer the question, it was assumed / deduced that the figure or shape that could give the largest area for the given perimeter is the circle.
Question: Is there a clue on the problem that states that the largest area that can be formed is through a circle? As there are other shapes the 20 m long bar can be formed. Also, we can't really tell if it's true as I think there is not enough information to answer that.
Sorry if the question seems odd. It's just so I can improve my visualization in this type of problems.
 A: The largest area that can be formed is  a circle.
The area of a circle is $A = \pi r^2$
You can find $r$ by using the perimeter formula $P = 2\pi r  = 20 \implies r= \dfrac{10}{\pi}$
So the area if a circle is made is $A = \dfrac{100}{\pi}\approx31.83 \;m^2$
Lets take regular $n$ - gons, the area of a regular $n-$gon is given by ;
$A = \dfrac14ns^2\cot\bigg(\dfrac \pi n\bigg)$
where $r$ is the distance from the center to the vertex of any regular polygon.
You can see that the perimeter $P = 20 = n\cdot s  $  $\; $ where $s$ is the length of one side of the polygon .
$\implies s= \dfrac{20}n$
$\therefore A  = \displaystyle \frac 14n\frac{400}{n^2}\cot\bigg(\frac\pi n\bigg) = \frac{100}n\cot\bigg(\frac \pi n\bigg)$
for $\begin{pmatrix}n&&&&&& A\\3&&&&&&19.24\\4&&&&&&25\\5&&&&&&27.527\\6&&&&&&28.867\\7&&&&&&29.6645\\8&&&&&&30.17767\\9&&&&&&30.5275\\\vdots\\\vdots\\99999&&&&&&31.83\end{pmatrix}$
The limiting case as $n\to \infty$ (Which is basically  a circle) gives the area $A = 31.83\; m^2$.
Thus we could conclude the largest area is the circle
EDIT: 
As user Aaron mentioned , this proof is not rigorous , but merely an intuitive approach to it. 
