Shapes play an important role in many human activities, but are rarely seen in their natural form as raw and unanalyzed. Rather, shapes come analyzed, structured in terms of their certain parts, forming shape decompositions. Different kinds of shape decompositions are developed, the most interesting among which are the decompositions that could be used as shape approximations. Two kinds of such decompositions, discrete and bounded, are examined in greater detail. Computations with shapes conducted in the framework of shape grammars and related shape algebras have been standard for over 3 decades. Similar computations are possible with analyzed shapes or shape decompositions. Different algebras to compute with shape decompositions are developed and compared to the shape algebras. The measure of their agreement determines how well the shapes are approximated by their decompositions.