  We introduce a method to estimate continuum percolation thresholds and
illustrate its usefulness by investigating geometric percolation of
non-interacting line segments and disks in two spatial dimensions. These
examples serve as models for electrical percolation of elongated and flat
nanofillers in thin film composites. While the standard contact volume argument
and extensions thereof in connectedness percolation theory yield accurate
predictions for slender nanofillers in three dimensions, they fail to do so in
two dimensions, making our test a stringent one. In fact, neither a systematic
order-by-order correction to the standard argument nor invoking the
connectedness version of the Percus-Yevick approximation yield significant
improvements for either type of particle. Making use of simple geometric
considerations, our new method predicts a percolation threshold of $\rho_c l^2
\approx 5.83$ for segments of length $l$, which is close to the $\rho_c l^2
\approx 5.64$ found in Monte Carlo simulations. For disks of area $a$ we find
$\rho_c a \approx 1.00$, close to the Monte Carlo result of $\rho_c a \approx
1.13$. We discuss the shortcomings of the conventional approaches and explain
how usage of the nearest-neighbor distribution in our new method bypasses those
complications.
