Consider the following heat conduction problem. Let D be an open, bounded and connected set in euclidean space R2 with a polygonal boundary. Suppose that D has temperature 1 at time t=0, while the boundary is kept at temperature 0 for all time t>0. We obtain the asymptotic behaviour for the amount of heat in D at time t up to O(e-q/t) as t?0. © 1990 Springer-Verlag.