The oriented strip packingproblem is very important to manufacturing industries: given a strip of fixed width and a set of many (> 100) nonconvex polygons with 1, 2, 4, or 8 orientations permitted for each polygon, find a set of translations and orientations for the polygons that places them without overlapping into the strip of minimum length. Heuristics are given for two versions of strip packing: 1) translation-only and 2) oriented. The first heuristic uses an algorithm we have previously developed for translational containment: given polygons P1, P2…., Pk and a fixed container C, find translations for the polygons that place them into C without overlapping. The containment algorithm is practica] for k ≤ 10. Two new containment algorithms are presented for use in the second packing heuristic. The first, an ordered containment algorithm, solves containment in time which is only linear in k when the polygons are a) "long" with respect to one dimension of the container and b) ordered with respect to the other dimension. The second algorithm solves compliant containment: given polygons P1, P2,..., Pi+k and a container C such that polygons P1, P2,…, Pi are already placed into C, find translations for Pi+l, Pi+2,…, Pi+k and a nonoverlapping translationa] motion of P1, P2,…, Pi that allows all l + k polygons to fit into the container without overlapping. The performance of the heuristics is compared to the performance of commercial software and/or human experts. The results demonstrate that fast containment algorithms for modest values of k (k ≤ 10) are very useful in the development of heuristics for oriented strip packing of many (k >> 10) polygons.