The first known exact solutions are derived for the free vibrations of thick (Mindlin) annular sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain non-integer order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. Frequency determinant equations are derived for thick annular sectorial plates with circular edges having all nine possible combinations of clamped, simply supported, or free boundary conditions. Extensive amounts of nondimensional frequency parameters are presented for thickness ratio ( h a) values of ≅ 0,0.1, and 0.2; radii ( b a) values within the range of 0.1-0.7; and sector angle values of 180° ≤ α ≤ 360° for which, in the range of α > 180°, no previously published results are known to exist. Frequency results obtained for thick annular sectorial plates are compared to those determined for classically thin ( h a ≅ 0) ones. The frequencies for 360° annular sectorial plates (i.e. annular plates having a hinged crack) are compared with those for complete circular annular plates. The exact solutions presented herein are useful to researchers for determining the correctness of approximate numerical procedures and software packages for thick plate vibration analyses.