A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a generic translation surface. We consider the stratum H_1(2g-2) of translation surfaces of genus g with one singularity and show that the expected diameter of a surface is bounded above by a uniform multiple of ((log g)/g)^(1/2). This is smaller than what one would expect by analogy from the result of Mirzakhani about the expected diameter of a hyperbolic metric on a Riemann surface. In fact, more generally, we compute the expected value of the covering radius of a translation surface in any stratum H_1(kappa). To prove our result, we need an estimate for the volume of the thin part of H_1(kappa) which is given in the appendix.