Manmade metamaterials are used due to their different properties originating from their geometry. Sound attenuation is a property that depends on structure geometry.  Small structures could reduce sound propagation within a mid-high frequency. To change the range of sound propagated in metamaterial from high-frequency to low-frequency, however, internal resonators could be used due to their rotational vibration having high effect on sound attenuation. In this paper, an acoustical analysis is done on a hexachiral lattice structure with an internal resonator capable of decreasing sound wave propagation among the structure in the low-frequency range, which causes corresponding bandgaps in this frequency range (1-4500 Hz). Geometry parameters that can affect the width and range of bandgaps are studied, including the radius of the resonator, the thickness of the ligament, and the distance between each node. It was found that the radius of the resonator has a positive impact on the ability of attenuation, but the distance between each node has a more negative impact on the bandgap. Optimization is done on the geometric parameters considering the weight of the structure, which leads to the construction of light structures capable of reducing sound propagation.