Consider a class R 1 of non-convex smooth obstacles B such that there exist a number of directions { s j , − s j }, j = 1, 2,…, N, uniformly distributed on the unit sphere with the following property: the rays incident in the directions +- s j are reflected by the obstacle once according to geometrical optics laws. It is also assumed that the obstacles in R 1, satisfy a non-trapping condition. For B ϵ R 1 the high frequency asymptotics of the scattering amplitude f( s, s 0, k) is studied and analytical formulas are derived for determining the surface of the obstacle from the scattering amplitude known at a fixed large wave number k, and for the directions s 0 and s of the incident and scattered waves in solid angles around the directions ± s j , j = 1,…, N. The investigation of the scattering amplitude and the solution to the inverse scattering problem are based on approximate formulas for f derived from the Kirchhoff approximation, and these formulas are justified for non-convex obstacles of class R 1.