Ion radii are derived from the characteristic (grand mean) bond-lengths reported by Gagné & Hawthorne (2016-2020) for ordered crystal structures. The radius of a cation may be derived from the corresponding observed bond length by subtracting the radius of the constituent anion. To do this, it is necessary to know the radius of the constituent anion. Atom radii may also be derived by quantum mechanical calculation. Extensive quantum mechanical calculations indicate that the bonded radii of both cations and anions vary widely depending on the individual bond-pair considered. There is a dichotomy between the experimental approach for deriving radii, which assumes a constant (or near-constant) radius for each individual anion, and the quantum mechanical approach, which calculates large variations in individual anion (and cation) radii. This problem may be resolved by considering the two categories of use for ion radii: (1) those methods which use the relative sizes of cation and anion radii to predict local atomic arrangements; (2) those methods which compare the radii of different cations (or the radii of different anions) to predict local atomic arrangements. There is much uncertainty with regard to the relative sizes of cations and anions, giving rise to the common failure of type (1) methods, e.g. Pauling's first rule which purports to relate the coordination adopted by cations to the radius ratio of the constituent cation and anion. Conversely, type (2) methods which involve comparing the sizes of different cations with each other (or different anions with each other) can give very accurate predictions of site occupancies, physical properties etc. Type (2) methods use ion radius as a proxy variable for characteristic (grand mean) bond-length and the value of the radius of the anion used to calculate the radii is irrelevant to these applications; one can equally well use the characteristic bond lengths themselves (from which the radii are derived) to develop type (2) relations. The dichotomy between the experimentally derived ion radii and the quantum-mechanical calculations of electron density in crystals is removed by the recognition that ion radii calculated from mean bond lengths do not represent the sizes of those ions in crystals, and do not work for type (1) relations. Ion radii are proxy variables for characteristic bond-lengths for type (2) relations.