This paper develops a systematic procedure for constructing an attainable region (AR). The approach uses two dimensional ARs constructed in orthogonal subspaces to construct higher dimensional ARs. Our technique relies on previous algorithms that provide a practical assurance of the completeness of ARs in two dimensions, using only PFR and CSTR reactors and mixing. Here we build on a modification of this property by constructing 2D projections and their intersections that provide upper and lower bounds of the AR. These bounds are then improved by applying AR constructions sequentially to candidate regions in orthogonal subspaces. The approach is demonstrated on a well-known AR problem in three dimensions.