A polygon in the hyperbolic plane is cyclic if a single circle contains all
of its vertices; we will say it is "centered" if in addition its interior
contains the center of this circle. We give necessary and sufficient conditions
for a set of real numbers to be the side length collection of a cyclic or
centered polygon. A cyclic polygon is uniquely determined by its collection of
side lengths; its vertex angles vary as C^1 functions of side lengths; and so
does the radius of the circle containing its vertices.