Many biological processes rely on the ability of cells to measure local ligand concentration. However, such measurements are constrained by noise arising from diffusion and the stochastic nature of receptor–ligand interactions. It is thus critical to understand how accurately, in principle, concentration measurements can be made. Previous theoretical work has mostly investigated this in 3D under the simplifying assumption of an unbounded domain of diffusion, but many biological problems involve 2D concentration measurement in bounded domains, for which diffusion behaves quite differently. Here we present a theory of the precision of chemosensation that covers bounded domains of any dimensionality. We find that the quality of chemosensation in lower dimensions is controlled by domain size, suggesting a general principle applicable to many biological systems. Applying the theory to biological problems in 2D shows that diffusion-limited signalling is an efficient mechanism on time scales consistent with behaviour.