The authors introduce the notion of ‘finite’ type iterated function systems of contractive similitudes to treat iterated function systems that do not satisfy the open set condition. This property leads to an explicit formula for the exact Hausdorff dimension of their attractors, which extends previous results of S. P. Lalley [Trans. Am. Math. Soc. 349, No. 11, 4355-4365 (1997; Zbl 0919.11053) ]. The result applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.