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
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 $3$.