We calculated and measured the density distribution and cloud size of a trapped two-dimensional (2D) 6Li Fermi gas near a Feshbach resonance at low temperatures. Density distributions and cloud sizes were calculated for a wide range of interaction parameters using a local density approximation (LDA) and a zero-temperature equation of state obtained from quantum Monte Carlo simulations reported by Bertaina and Giorgini (2011 Phys. Rev. Lett. 106 110403). We found that LDA predictions agree well with experimental measurements across a Feshbach resonance. Theoretical results for Tan's contact parameter in a trapped gas are reported here along with predictions for the static structure factor at large momentum which could be measured in future Bragg spectroscopy experiments on 2D Fermi gases.