Schwarzian field theory has played an important role in recent studies of the SYK model and JT gravity. A rigorous construction in terms of random circle reparametrisations was recently put forward by Bauerschmidt, Losev and Wildemann. In this talk I will show that it is also obtained as a scaling limit of a natural model, introduced by Chekhov, of random ideal polygons in the hyperbolic plane. Finally, I will comment on its role within the broader landscape of random planar geometry.