The paper studies the relation between the asymptotic values of the ratios area/length (F/L) and diameter/length (D/L) of a sequence of convex sets expanding over the whole hyperbolic plane. It is known that F/L goes to a value between 0 and 1 depending on the shape of the contour. In the paper, it is first of all seen that D/L has limit value between 0 and 1/2 in strong contrast with the euclidean situation in which the lower bound is 1/π (D/L = 1/π if and only if the convex set has constant width). Moreover, it is shown that, as the limit of D/L approaches 1/2, the possible limit values of F/L reduce. Examples of all possible limits F/L and D/L are given. © 2001 London Mathematical Society.