The classical, in its nature, concept of atomic or ionic radii, although profitable in many fields, is represented by an ambiguous choice of formulations. In this work, we propose a definition of atomic and ionic radii rooted in chemical principles and conceptual density functional theories. The estimation based on electron density fundamental response functions has been successfully tested. The generalized approach has been shown to be applicable to atoms in any oxidation state. The radii display good correlation with classical estimates, such as Shannon. The atomic and ionic radii obtained according to this scheme are directly comparable between different elements, without any adjustment procedures requiring fitting constants. The definition also has a clear physical interpretation, which supports understanding of size-related phenomena and trends.