  We consider $N$ circles of equal radii, $r$, having their centers randomly
placed within a square domain $\mathcal{D}$ of size $L \times L$ with periodic
boundary conditions ($\mathcal{D} \in \mathbb{R}^2$). When two or more circles
intersect each other, each circle is divided by the intersection points into
several arcs. We found the exact length distribution of the arcs. In the
limiting case of dense systems and large size of the domain $\mathcal{D}$ ($L
\to \infty$ in such a way that the number of circle per unit area, $n=N/L^2$,
is constant), the arc distribution approaches the probability density function
(PDF) $f(\psi) = 4 n r^2\exp(-4 n r^2 \psi)$, where $\psi$ is the central angle
subtended by the arc. This PDF is then used to estimate the sheet resistance of
transparent electrodes based on conductive rings randomly placed onto a
transparent insulating film.
