  We explicitly compute the limiting gap distribution for slopes of saddle
connections on the flat surface associated to the regular octagon with opposite
sides identified. This is the first such computation where the Veech group of
the translation surface has multiple cusps. We also show how to parametrize a
Poincar\'e section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,\omega)$
associated to an arbitrary Veech surface $(X, \omega)$. As a corollary, we show
that the associated gap distribution is piecewise real analytic.
