The full width at half maximum (FWHM) of the Voigt profile can be found from the
widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile 
is
The FWHM of the Lorentzian profile is
An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is:
By construction, this expression is exact for a pure Gaussian or Lorentzian.
A better approximation with an accuracy of 0.02% is given by  (originally found by Kielkopf)
Again, this expression is exact for a pure Gaussian or Lorentzian.
In the same publication, a slightly more precise (within 0.012%), yet significantly more complicated expression can be found.