An abelian differential equips a Riemann surface with a singular flat structure. Unlike the hyperbolic metric on a Riemann surface, not much is known about the shape of a generic singular flat surface. We consider the stratum $H_1(2g-2)$ of abelian differentials on a surface of genus g with one zero. We show that the average diameter of a surface in $H_1(2g-2)$ goes to zero as g goes to infinity at a rate of $1/g^{1/2-\epsilon}$. In contrast, the minimum possible diameter of a surface in $H_1 (2g-2)$ goes to zero at a rate of $1/g^{1/2}$. This is joint work with Howard Masur and Anja Randecker.