This paper introduces a new solid, four-node, fully parameterized, tetrahedron element. The tetrahedron element is based on the absolute nodal coordinate formulations in which the absolute position vector and three slope vectors are considered nodal coordinates. The linear transformation of the shape functions from the tetrahedron coordinates to Cartesian coordinates leads to very stiff elements. On the other hand, the nonlinear transformation from tetrahedron coordinates to Cartesian coordinates is very complicated. Therefore, the use of the barycentric coordinates was bypassed and the Cartesian coordinates were used to define the cubic polynomial shape functions. The volumetric integration of the mass matrix, elastic force and gravity force was performed using the trapezoidal rule. It is easily shown that the resulting element satisfies the conformity conditions. The numerical results show that the proposed element does not suffer from Poisson locking. Furthermore, the numerical results show that comparing to fully parametrized brick elements, the proposed element experiences poor convergency and excessive computational time.