We consider the Cauchy problem for a semilinear heat equation with 
a supercritical power nonlinearity. It is known that the asymptotic behavior
of solutions in time is determined by the decay rate of their initial values in
space.  In particular, if an initial value decays like a radial steady state, then
the corresponding solution converges to that steady state.  In this paper we
consider solutions whose initial values decay in an anisotropic way. We
show that each such solution converges to a steady state which is explicitly
determined by an average formula. The proof is given by using our previous
results on global stability and quasi-convergence of solutions, self-similar
solutions of the linearized equation around a singular steady state, and
a comparison technique.