the radius ratio
The radius ratio is the ratio of the smaller ion's radius to the larger ion's radius, usually written as the cation radius over the anion radius. It is a simple geometric idea with a big payoff: it predicts how many large ions can pack around a small one, and therefore which structure an ionic compound will adopt. Fit a small ball among big balls and only so many big ones can touch it at once.
The geometry gives clean thresholds. Below about 0.155 the cation prefers 2 or 3 neighbours; from 0.155 to 0.225 it takes 3 (triangular); from 0.225 to 0.414 it fits a tetrahedral hole with 4 neighbours; from 0.414 to 0.732 an octahedral hole with 6; and from 0.732 to 1.0 a cubic site with 8. Worked example: sodium chloride has r+ near 102 pm and r- near 181 pm, so the ratio is 0.56, which lands in the octahedral range and predicts coordination 6, exactly the rock-salt structure. Caesium chloride has 167 over 181, a ratio of 0.92, which predicts coordination 8 and gives the caesium-chloride structure.
Treat the radius ratio as a helpful guideline, not an iron law. It assumes ions are hard, undeformable spheres in purely ionic bonding, and it ignores covalent character and polarisability, so it predicts the right coordination only about two-thirds of the time. Zinc sulfide, for example, is fourfold coordinated even though its radius ratio suggests six, because its bonding is strongly covalent.
Na+/Cl- = 102/181 = 0.56 predicts octahedral coordination 6 (rock salt); Cs+/Cl- = 167/181 = 0.92 predicts cubic coordination 8 (CsCl).
Bigger cation relative to anion, more neighbours it can support.
The radius ratio rule fails when bonding is covalent or ions are polarisable, and ionic radii themselves depend on coordination, so use it to guess, then check against the real structure.