Estimating pole radius of formant filter from bandwidth
I have here a very pretty answer supplied by @Jazzmaniac, which I will transcribe from our conversation (on IRC Freenode ##dsp).



The key insight is that the gain is inversely proportional to the distance from the pole.

So, double the distance, halve the gain.

(This can be seen by imagining a single pole on the origin with the transfer function $H(z) = 1/z$)

The small circle represents the peak gain (at frequency $e^{jθ}$). 

The bigger circle represents a locus of half that gain.

Hence, it is clear that Φ = bandwidth/2