  In this paper, we are interested in flat metric structures with conical
singularities on surfaces which are obtained by deforming translation surface
structures. The moduli space of such flat metric structures can be viewed as
some deformation of the moduli space of translation surfaces. Using geodesic
triangulations, we define a volume form on this moduli space, and show that, in
the well-known cases, this volume form agrees with usual ones, up to a
multiplicative constant.
