A low-density structured material with good mechanical stability that can be used for three-dimensional structures, and methods to make and use same. In embodiments, the low-density structured material includes a first surface of interconnected polyhedrons, a plurality of tetrahedral arrangements whose base is the polyhedrons of the first surface, a second surface that is a web attached to the tetrahedral vertices of the tetrahedral arrangements, and panel materials overlying the web. The low-density structured material can be utilized in a variety of different structures.