Q:

Given the area of $1000$ sided polygon ,how could we find the  maximum length  between any two of its vertices?

If  the area of $1000$ sided polygon is $314$ sq. cms. How to find
  the  maximum length (in cm) between any two vertices of this polygon?
$
a)\space 15.7 \quad   \quad b)\space 18.1  \quad \quad c)\space 20  \quad\quad d) \space 21.6$

I am not sure how to solve this for a general polygon.Any ideas?
$\lt$Problem Source $\gt$

A:

Just for the reason,that this question is still unanswered,I am posting this official solution from here,which is very close to Matt Bennet's hint in the comments above.

It can be seen that, the higher the number of sides of a regular
  polygon, the more closely does its area approach to that of its
  circum-circle. 
In this case, we have a polygon of $1000$ sides and its area will be
  very close to that of the circle of radius $r$.
To find $r$, we put,
$πr² = 314$ cm$^²$
$\Rightarrow r ≈ 10$ cm
Now, vertices $1$ and $501$ of our $1000$ sided polygon will
  correspond to the opposite ends of the diameter of the circum-circle
  of this polygon.
∴ The distance between them will be approximately = $2 × r = 20$ cm
Hence, option $3$.

