A Brooks Coil is generally known as a special case of the circular coil inductor of rectangular cross section that is implemented to achieve the maximum inductance with a given length of wire. As can be seen from the picture below, the coil has a square cross section and the inner diameter is equal to twice the height (or width) of the coil winding.
The inductance for a Brooks coil can be found from the following equation:
where c is the height and width of the coil winding (in cm) and N is the number of turns. A second formula is shown below (Grover, Frederick W., Inductance Calculations Working Formulas and Tables, D. Van Nostrand Company, 1946, p98. ISBN:0486495779):
where a is the mean radius of the inductor (in cm) and N is the number of turns.