The first approximation is based on a modification of an expression developed by Wheeler; the second is derived from electromagnetic principles by approximating the sides of the spirals as current-sheets; and the third is a monomial expression derived from fitting to a large database of inductors (and the exact inductance values).
All three expressions are accurate, with typical errors of 2 – 3%, and very simple, and are therefore excellent candidates for use in design and synthesis.The thickness of the inductor has only a very small effect on inductance and will therefore be ignored.
More about planar spiral air core inductors
In the same category as the flat spiral coils we have planar spiral coils, planar square spiral coils, planar rectangular spiral coils, planar hexagonal spiral coils and octagonal spiral coil. Planar coils are mostly used in high frequency applications and designed as tracks on a circuit board.
What is an air core inductor?
An “air core inductor” is an inductor that does not depend upon a ferromagnetic material to achieve its specified inductance. Some inductors are wound without a bobbin and just air as the core. Some others are wound on a bobbin made of bakelite, platsic, ceramic etc.
For more detail: Single layer Planar spiral coil inductor calculator
French 
English