  We study black hole solutions in $R^4\times S^1$ space, using an expansion to
fourth order in the ratio of the radius of the horizon, $\mu$, and the
circumference of the compact dimension, $L$. A study of geometric and
thermodynamic properties indicates that the black hole fills the space in the
compact dimension at $\epsilon(\mu/L)^2\simeq0.1$. At the same value of
$\epsilon$ the entropies of the uniform black string and of the black hole are
approximately equal.
