  Diameter is one of the most basic properties of a geometric object, while
Riemann surfaces are one of the most basic geometric objects. Surprisingly, the
diameter of compact Riemann surfaces is known exactly only for the sphere and
the torus. For higher genuses, only very general but loose upper and lower
bounds are available. The problem of calculating the diameter exactly has been
intractable since there is no simple expression for the distance between a pair
of points on a high-genus surface. Here we prove that the diameters of a class
of simple Riemann surfaces known as generalized Bolza surfaces of any genus
greater than $1$ are equal to the radii of their fundamental polygons. This is
the first exact result for the diameter of a compact hyperbolic manifold.
