I think it is more a matter of definition than a derivation.

If we assume a perfect, unstrained cubic perovskite unit cell, and try to find the relation between the ionic radii and the lattice parameter a, then on one hand, going through the center B ion:

a=2rO+2rB

and on the side wall, we can use the diagonal:

a=(2rA+2rO)/20.5

In practice you will seldom find a perfect match between these three radii. The deviation from this ideal case can be defined as the ratio between these two equations, also known as the Goldschmidt tolerance factor

Dear Lior Kornblum,

Thanks lot for your valuable answer.

To complete the answer, you can find the ionoc radii on the following page:

http://abulafia.mt.ic.ac.uk/shannon/ptable.php

hoping it helps

G. Dezanneau

is tolerance factor equation is same for any perovskite structure. can anyone give information about the calculation of tolerance factor for AA'B2O5+δ structure