We consider the short-time transient of the diffusion of mass (or heat) to a body whose surface is held at zero concentration, with the space outside initially at unit concentration. The problem is expressed in terms of the probability that a brownian path of short duration intersects the body. A three-term asymptotic series for the absorption rate is derived for an arbitrary smooth body, together with the leading corrections due to edges and lines of contact with insulating surfaces. A three-term series is also derived for a plane laminar conductor with a smooth boundary, or equivalently a conducting film mounted on an insulating plane. These results are used to derive short-time absorption rates for shapes such as discs, rings and cylinders, commonly used for microelectrodes and hot-film devices.