A common realization of superhydrophobic surfaces makes use of a mesh-like geometry, where pockets of air are trapped in a periodic array of holes in a no-slip solid substrate. We consider the small-solid-fraction limit where the ribs of the mesh are narrow. In this limit, we obtain a simple leading-order approximation for the slip-length tensor of an arbitrary mesh geometry. This approximation scales as the solid-fraction logarithm, as anticipated by Ybert et al. [Phys. Fluids 19, 123601 (2007)]; in the special case of a square mesh it agrees with the analytical results obtained by Davis & Lauga [Phys. Fluids 21, 113101 (2009)].