Abstract: Lattice structures have drawn much attention in engineering applications due to their lightweight and multi-functional properties. In this work, a mathematical design approach for functionally graded (FG) and helicoidal lattice structures with triply periodic minimal surfaces is proposed. Four types of lattice structures including uniform, helicoidal, FG, and combined FG and helicoidal are fabricated by the additive manufacturing technology. The deformation behaviors, mechanical properties, energy absorption, and acoustic properties of lattice samples are thoroughly investigated. The load-bearing capability of helicoidal lattice samples is gradually improved in the plateau stage, leading to the plateau stress and total energy absorption improved by over 26.9% and 21.2% compared to the uniform sample, respectively. This phenomenon was attributed to the helicoidal design reduces the gap in unit cells and enhances fracture resistance. For acoustic properties, the design of helicoidal reduces the resonance frequency and improves the peak of absorption coefficient, while the FG design mainly influences the peak of absorption coefficient. Across broad range of frequency from 1000 to 6300 Hz, the maximum value of absorption coefficient is improved by 18.6%–30%, and the number of points higher than 0.6 increased by 55.2%–61.7% by combining the FG and helicoidal designs. This study provides a novel strategy to simultaneously improve energy absorption and sound absorption properties by controlling the internal architecture of lattice structures.