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The mechanical properties of tissues are influenced by those of constituent cells in various ways. For instance, it has been theoretically demonstrated that two-dimensional confluent tissues comprising mechanically uniform cells can undergo density-independent rigidity transitions, and analysis of the dynamical behavior of tissues near the critical point revealed that the transitions are geometrically controlled by the so-called cell shape parameter. To investigate whether three-dimensional tissues behave similarly to two-dimensional ones, we herein extend the previously developed model to three dimensions both dynamically and statically, demonstrating that two mechanical states similar to those of glassy materials exist in the three-dimensional case. Scaling analysis is applied to the static model focused from the rearrangement viewpoint. The obtained results suggest that the upper critical dimension of tissues equals two and is therefore the same as that of the jamming transition.