  Tutte proved that every 4-connected planar graph contains a Hamilton cycle,
but there are 3-connected $n$-vertex planar graphs whose longest cycles have
length $\Theta(n^{\log_32})$. On the other hand, Jackson and Wormald in 1992
proved that an essentially 4-connected $n$-vertex planar graph contains a cycle
of length at least $(2n+4)/5$, which was recently improved to $5(n+2)/8$ by
Fabrici {\it et al}. In this paper, we improve this bound to $\lceil
(2n+6)/3\rceil$ for $n\ge 6$, which is best possible, by proving a quantitative
version of a result of Thomassen on Tutte paths.
