Let $A_n$ be the minimum possible largest area inside a unit circle traced by $n$ lines.
A circle can be divided into $\frac{n^2+n+2}2$ regions by tracing $n$ chords, so $A_n\geq\frac{2\pi}{n^2+n+2}$ is an easy lower bound. By making the chords all evenly spaced diameters we get $A_n\leq \frac \pi {2n}$ but I get the feeling that this is very loose and $A_n$ will be closer to the lower bound for large n.
For $n=1,2$ the lower bound is easily achieved, but for $n=3$ it is difficult to beat the upper bound $\frac\pi 6$.
In a symmetrical arrangement like this the area of the three large outside sections is always $\geq\pi/6$, so if there's a better arrangement it must not have triangular symmetry. In fact I think this is the case for all $n\geq 3$.
So what can be said about $A_n$? I conjecture that
$$\lim_{n\to\infty}n^2A_n<\infty$$
But beyond that I find it difficult to determine anything about it.
By taking $n/2$ equally spaced horizontal and $n/2$ equally spaced vertical lines, the regions all have area $\le (2/(n/2+1))^2= O(1/n^2)$.