OK, a third formula now. If anybody is still keeping score, then Fat Dog's
formula remains exact (modulo all the simplifying assumptions Tim Zukas
enumerates), keendog's formula is a good approximation, and
Tim Zukas's approximation very nearly splits the difference between
the two dogs' answers.
Since 6080/6886 is about 0.883 the conversion to nm is X^2/6886.
Now if one expands the answers of the two dogs in a
Maclaurin series
(which in this case is just Newton's Binomial Formula)
one finds the simple quadratic approximation X^2/6886 to be
nearly midway between the two earlier answers. The
plot
shows that all three answers are very nearly the same for X < 200 nm.
(Just thought I'd make all this clear.)