We consider the measure of the set of all unit vectors tangent to rays emanating from a point p p in a finitely connected complete open Riemannian 2 2 -manifold M M . If M M with one end admits total curvature c ( M ) c(M) , then this measure tends to min { 2 π χ ( M ) − c ( M ) , 2 π } \min \{ 2\pi \chi (M) - c(M),2\pi \} as p p tends to infinity, where χ ( M ) \chi (M) is the Euler characteristic.