  We investigate the heat flow in an open, bounded set $D$ in $\mathbb{R}^2$
with polygonal boundary $\partial D$. We suppose that $D$ contains an open,
bounded set $\widetilde{D}$ with polygonal boundary $\partial \widetilde{D}$.
The initial condition is the indicator function of $\widetilde{D}$ and we
impose a Neumann boundary condition on the edges of $\partial D$. We obtain an
asymptotic formula for the heat content of $\widetilde{D}$ in $D$ as time
$t\downarrow 0$.
