  We compute the $F$-pure threshold of non necessarily principal ideals which
satisfy a geometric generic condition about their Newton polygons. We also
contribute some evidence in favor of the conjectured equality between the
$F$-pure threshold and the log canonical threshold of ideals. These results are
obtained by generalizing the theory of splitting polytopes to the case of
ideals. As applications of our results we obtain geometric lower bounds for the
recently introduced $F$-volume of a collection of ideals.
