  Triangulated surfaces are compact Riemann surfaces equipped with a conformal
triangulation by equilateral triangles. In 2004, Brooks and Makover asked how
triangulated surfaces are distributed in the moduli space of Riemann surfaces
as the genus tends to infinity. Mirzakhani raised this question in her 2010 ICM
address. We show that in the large genus case, triangulated surfaces are well
distributed in moduli space in a fairly strong sense. We do this by proving
upper and lower bounds for the number of triangulated surfaces lying in a
Teichm\"uller ball in moduli space. In particular, we show that the number of
triangulated surfaces lying in a Teichm\"uller unit ball is at most exponential
in the number of triangles, independent of the genus.
