•A cell-based smoothed finite element method is proposed to make a polygonal element for analyzing FSI problems.•A sliding polygonal mesh method is developed to prevent mesh entanglement in the fluid domain.•The present scheme provides a seamless connection between dissimilar meshes at the interface.•The present scheme yields better accuracy compared with the conventional FEM. A cell-based smoothed finite element method is proposed to make a polygonal type variable-node element for analyzing two-dimensional FSI problems. The gradient smoothing technique for the velocity field provides a proper softening effect, which relieves the stiff behavior of the conventional FEM. In addition, by using polygonal elements, a seamless connection is achieved between non-matching meshes satisfying the interface conditions of continuity and compatibility. In order to prevent mesh distortion along the fluid-solid interface, a sliding polygonal mesh was developed using the cell-based smoothed finite element method. Some numerical examples were presented to assess the performance and the effectiveness of the present method. The smoothed finite element methods yielded better accuracy compared with the conventional FEM, and the sliding polygonal mesh method successfully treated mesh distortion.