We give several necessary and sufficient conditions for an $AH$ algebra to have its ideals generated by their projections. Denote by ${\mathcal C}$ the class of $AH$ algebras as above and in addition with slow dimension growth. We completely classify the algebras in ${\mathcal C}$ up to a shape equivalence by a $K$-theoretical invariant. For this, we show first, in particular, that any $C^{ * }$-algebra in ${\mathcal C}$ is shape equivalent to an $AH$ algebra with slow dimension growth and real rank zero (generalizing so a result of Elliott-Gong); then, we use a classification result of Dadarlat-Gong. We prove that any $AH$ algebra in ${\mathcal C}$ has stable rank one (i.e., in the unital case, that the set of the invertible elements is dense in the algebra), generalizing results of Blackadar-Dadarlat-R\o rdam and of Elliott-Gong. Other nonstable $K$-theoretical results for $C^{ * }$-algebras in ${\mathcal C}$ are also proved, generalizing results of Dadarlat-N\'{e}methi, Martin-Pasnicu and Blackadar.