  We compute higher moments of the Siegel--Veech transform over quotients of
$SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization
of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear
algebra to create explicit integration formulas which give information about
densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which
arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt
on the variance of the Siegel transform over
$SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
