For rotationally invariant first passage percolation (FPP) on the plane, we use a multi-scale argument to prove stretched exponential concentration of the first passage times at the scale of the standard deviation. Our results are proved under hypotheses which can be verified for many standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. This is the first such tight concentration result known for any model that is not exactly solvable. As a consequence, we prove a version of the so called KPZ relation between the passage time fluctuations and the transversal fluctuations of geodesics as well as up to constant upper and lower bounds for the non-random fluctuations in these models. Similar results have previously been known conditionally under unproven hypotheses, but our results are the first ones that apply to some specific FPP models. Our arguments are expected to be useful in proving a number of other estimates which were hitherto only known conditionally or for exactly solvable models.