Given a chemical reaction network (with kinetics) and given a specified feed, we ask which product compositions are attainable from the feed by means of processes that employ only reaction and mixing. Although there has been considerable attention to this question recently, much of the "attainable region" research has focused on building the set of realizable compositions "from the inside", that is, by attempting to extend a set of already realized compositions with further reaction and mixing. Here we seek to bound the set of attainable compositions "from the outside" by construction of a bounding polygon in composition space, within which all attainable compositions must reside. The polygon construction proceeds by what we call the method of bounding hyperplanes. When the number of hyperplanes employed becomes large, the resulting polygonal bound can, at least in some instances, approximate the actual attainable region sharply.