Agol’s veering triangulation for 3-manifolds that fiber over the circle can be obtained very explicitly, via a construction of Gueritaud, from the stable and unstable foliations of the monodromy. We study the way in which these triangulations interact with the arc complexes of the surface and its subsurfaces. This allows us to examine the “profile” of subsurface projections associated to each fiber in a fibred face of the Thurston norm ball, finding explicit relations between these projections and the simplicial structure of the triangulations and obtaining some bounds that do not depend on the complexity of the fibers. Joint work with Sam Taylor