We have investigated the optical thickness of the coronal plasma through the analysis of high-resolution X-ray spectra of a large sample of active stars observed with the High Energy Transmission Grating Spectrometer on Chandra. In particular, we probed for the presence of significant resonant scattering in the strong Lyman series lines arising from hydrogen-like oxygen and neon ions. The active RS CVn-type binaries II Peg and IM Peg and the single M dwarf EV Lac show significant optical depth. For these active coronae, the Lyα/Lyβ ratios are significantly depleted as compared with theoretical predictions and with the same ratios observed in similar active stars. Interpreting these decrements in terms of resonance scattering of line photons out of the line of sight, we are able to derive an estimate for the typical size of coronal structures, and from these we also derive estimates of coronal filling factors. For all three sources we find that both the photon path length as a fraction of the stellar radius and the implied surface filling factors are very small and amount to a few percent at most. The measured Lyα/Lyβ ratios are in good agreement with APED theoretical predictions, thus indicating negligible optical depth, for the other sources in our sample. We discuss the implications for coronal structuring and heating flux requirements. For the stellar sample as a whole, the data suggest increasing quenching of Lyα relative to Lyβ as a function of both LX/Lbol and the density-sensitive Mg XI forbidden-to-intercombination line ratio, as might generally be expected.