In recent years, a new type of topological insulator, termed higher-order topological insulator, has attracted tremendous research interest. Such exotic lower-dimensional topological boundary states have been extended and reproduced in classical systems, such as optics and acoustics. In this paper, a two-dimensional acoustic honeycomb structure with a triangle resonant cavity is numerically studied. Topological phase transition is induced by gradually adjusting the intracell and intercell coupling, and then the topological phase is used to construct a second-order topological insulator. The topological properties of second-order topological insulators can be characterized by using the quantized quadrupole moments. When quantized quadrupole Q_ij = 0 , the system is trivial, while Q_ij = 1/2 , the system is topologically nontrivial. We investigate the acoustical higher-order states of triangular and hexagonal structures, respectively. The gapped zero-dimensional corner states are observed in both structures, but the robustness properties of the corner states emerge only in the hexagonal structures but not in the triangular-shaped ones. The topological corner modes will offer a new way to robustly confine the sound in a compact acoustic system.