Increasing efforts have recently been directed towards the question of how to incorporate the idea of a multidimensional poverty concept into traditional poverty measurement. In response, several suggestions have been made to derive different classes of multidimensional poverty measures. In this paper we focus on five axiomatically derived classes of multidimensional poverty measures. Each of these classes follow the unidimensional approach to progressively weight the respective distances to the threshold levels in order to account for poverty intensity. In this paper we claim that this approach, though reasonable in a unidimensional setting, does not suffice in a multidimensional setting. An additional aspect of poverty intensity should be considered which we denote as dimensional poverty: the number of dimensions in which individuals are deprived. There exists no luminous explanation why a weighting scheme should account for one aspect of poverty intensity while at the same time ignoring the other one. In this paper we introduce a multiple cutoff method to identify the poor which allows us to extent the five classes of poverty measures to include an additional weighting scheme in order to account for dimensional poverty. We find that the additional weight has no effect on the axiomatic basis of the classes of poverty measures other than a partial violation of the well-known subgroup decomposability axiom.