  The Farey sequence is a natural exhaustion of the set of rational numbers
between 0 and 1 by finite lists. Ford Circles are a natural family of mutually
tangent circles associated to Farey fractions: they are an important object of
study in the geometry of numbers and hyperbolic geometry. We define two
sequences of polygons associated to these objects, the Euclidean and hyperbolic
Farey-Ford polygons. We study the asymptotic behavior of these polygons by
exploring various geometric properties such as (but not limited to) areas,
length and slopes of sides, and angles between sides.
