Analytic approximation to 5 dimensional Black Holes with one compact dimension

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 $ϵ(\mu \mathrm{/}L{)}^2\simeq 0.1$. At the same value of $ϵ$ the entropies of the uniform black string and of the black hole are approximately equal.