Multi
layer air core inductor calculator
Formula used in this calculation is from Wheelers approximations
which is accurate to <1% if the cross section is
near square shaped.:

L (uH) =31.6*N^2* r1^2 / 6*r1+ 9*L + 10*(r2-r1)

where....
L(uH)= Inductance in microHenries
N = Total Number of turns
r1 = Radius of the inside of the coil in meters
r2 = Radius of the outside of the coil in meters
L = Length of the coil in meters

This formula applies at 'low' frequencies (<3MHz)
using enameled copper wire tightly wound.

More about air core inductors
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, plastic, ceramic etc.

Advantages of an air core coil:
Its inductance is unaffected by the current it carries.
This contrasts with the situation with coils using ferromagnetic
cores whose inductance tends to reach a peak at moderate
field strengths before dropping towards zero as saturation
approaches. Sometimes non-linearity in the magnetization
curve can be tolerated; for example in switching power
supplies and in some switching topologies this is an
advantage.
In circuits such as audio cross over filters in hi-fi
speaker systems you must avoid distortion; then an air
coil is a good choice. Most radio transmitters rely
on air coils to prevent the production of harmonics.
Air coils are also free of the "iron losses"
which a problem with ferromagnetic cores. As frequency
is increased this advantage becomes progressively more
important. You obtain better Q-factor, greater efficiency,
greater power handling, and less distortion.
Lastly, air coils can be designed to perform at frequencies
as high as 1 GHz. Most ferromagnetic cores tend to be
rather lossy above 100 MHz.

And the "downside":
Without a high permeability core you must have more
and/or larger turns to achieve a given inductance value.
More turns means larger coils, lower self-resonance
due to higher interwinding capacitance and higher copper
loss. At higher frequencies you generally don't need
high inductance, so this is then less of a problem.
Greater stray field radiation and pickup:
With the closed magnetic paths used in cored inductors
radiation is much less serious. As the diameter increases
towards a wavelength (lambda = c / f), loss due to electromagnetic
radiation will become significant. You may be able to
reduce this problem by enclosing the coil in a screen,
or by mounting it at right angles to other coils it
may be coupling with.
You may be using an air cored coil not because you require
a circuit element with a specific inductance per se
but because your coil is used as a proximity sensor,
loop antenna, induction heater, Tesla coil, electromagnet,
magnetometer head or deflection yoke etc. Then an external
radiated field may be what you want.

Brooks coil:
An interesting problem is to find the maximum inductance
with a given length of wire. Brooks, who wrote a paper
in 1931, calculated that the ideal value for the mean
radius is very close to 3A/2. As can be seen from the
picture below, the coil has a square cross section (A=B)
and the inner diameter is equal to twice the height
(or width) of the coil winding.
We call a coil having these dimensions a Brooks coil.
Brooks ratio is not critical. You can have a coil which
deviates from it quite significantly before the inductance
falls off too much. Also, you may have other considerations
than the inductance alone.

The inductance for a Brooks coil can be found from
the following equation:

L(uH)=0,025491*A*N^2

where A is the height and width of the coil winding
(in cm) and N is the number of turns. A second formula
is shown below:

L(uH)=0,016994*r*N^2

where r is the mean radius of the inductor (in cm)
and N is the number of turns.
(r=mean length of the coil radius measured from the
center of the coil to the center of the coil height,
as shown in the figure above.)