  Through inhalation of, e.g., hyperpolarized $^3$He, it is possible to acquire
gas diffusion magnetic resonance measurements that depend on the local geometry
in the vast network of microscopic airways that form the respiratory zone of
the human lung. Here, we demonstrate that this can be used to determine the
dimensions (length and radius) of these airways noninvasively. Specifically,
the above technique allows measurement of the weighted time-dependent diffusion
coefficient (also called the apparent diffusion coefficient), which we here
derive in analytic form using symmetries in the airway network. Agreement with
experiment is found for the full span of published hyperpolarized $^3$He
diffusion magnetic resonance measurements (diffusion times from milliseconds to
seconds) and published invasive airway dimension measurements.
