Polyhedra are widely used in art, science and technology. Faced with this situation, the following research question is formulated: Can new polyhedral structures be generated from another mathematical object such as a lemniscatic torus? For this question, the results we obtained are two particular cases whose vertices are points that belong to curves that lie on a lemniscatic torus: the first, a new polyhedron that has regular trapezoids in the equatorial zone, and the second, one that has triangles equal to each other. For both polyhedra, there exists an antipodal symmetry in the Arctic and Antarctic zones. Emphasis is placed on the construction of two convex polyhedra above mentioned: a one with 18-faces and other with 36-faces, using the scientific software Mathematica v.11.2. We also determine their total areas which respectively approximate 9.51 R2 and 10.44 R2. Likewise, the volume of each one is approximately 2.41 R3 and 4.19 R3, respectively. Moreover, they being inscribed in a sphere of radius R, and their opposite faces are not parallel.