We study the problem of decomposing (i.e. partitioning and covering) polygons into components that are $α$-fat, which means that the aspect ratio of each subpolygon is at most $α$. We consider decompositions without Steiner points. We present a polynomial-time algorithm for simple polygons that finds the minimum $α$ such that an $α$-fat partition exists. Furthermore, we show that finding an $α$-fat partition or covering with minimum cardinality is NP-hard for polygons with holes.