For typical ionic solids, the cations are smaller than the anions, and each cation is surrounded by coordinated anions which form a polyhedron. The sum of the ionic radii determines the cation-anion distance, while the cation-anion radius ratio  (or ) determines the coordination number (C.N.) of the cation, as well as the shape of the coordinated polyhedron of anions.
For the coordination numbers and corresponding polyhedra in the table below, Pauling mathematically derived the minimum radius ratio for which the cation is in contact with the given number of anions (considering the ions as rigid spheres). If the cation is smaller, it will not be in contact with the anions which results in instability leading to a lower coordination number.
The three diagrams at right correspond to octahedral coordination with a coordination number of six: four anions in the plane of the diagrams, and two (not shown) above and below this plane. The central diagram shows the minimal radius ratio. The cation and any two anions form a right triangle, with , or . Then . Similar geometrical proofs yield the minimum radius ratios for the highly symmetrical cases C.N. = 3, 4 and 8.
For C.N. = 6 and a radius ratio greater than the minimum, the crystal is more stable since the cation is still in contact with six anions, but the anions are further from each other so that their mutual repulsion is reduced. An octahedron may then form with a radius ratio greater than or equal to 0.414, but as the ratio rises above 0.732, a cubic geometry becomes more stable. This explains why  in NaCl with a radius ratio of 0.55 has octahedral coordination, whereas  in CsCl with a radius ratio of 0.93 has cubic coordination.
If the radius ratio is less than the minimum, two anions will tend to depart and the remaining four will rearrange into a tetrahedral geometry where they are all in contact with the cation.
The radius ratio rules are a first approximation which have some success in predicting coordination numbers, but many exceptions do exist. In a set of over 5000 oxides, only 66% of coordination environments agree with Pauling's first rule. Oxides formed with alkali or alkali-earth metal cations that contain multiple cation coordinations are common deviations from this rule.