  In this article, we study geometric aspects of semi-arithmetic Riemann
surfaces by means of number theory and hyperbolic geometry. First, we show the
existence of infinitely many semi-arithmetic Riemann surfaces of various shapes
and prove that their systoles are dense in the positive real numbers.
Furthermore, this leads to a construction, for each genus $g \geq 2,$ of
infinite families of semi-arithmetic surfaces with pairwise distinct invariant
trace fields, giving a negative answer to a conjecture of B. Jeon. Finally, for
any semi-arithmetic surface we find a sequence of congruence coverings with
logarithmic systolic growth and, for the special case of surfaces admitting
modular embedding, we are able to exhibit explicit constants.
