Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is 0.414...

Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is 0.414. (Hint: Use the NaCl crystal structure in Figure 12.2, and assume that anions and cations are just touching along cube edges and across face diagonals.)

Figure 12.2 A unit cell for the rock salt, or sodium chloride (NaCl), crystal structure.