  The purpose of this paper is twofold. First, we introduce a geometric
approach to study the circular orbit of a particle in static and spherically
symmetric spacetime based on Jacobi metric. Second, we apply the circular orbit
to study the weak gravitational deflection of null and time-like particles
based on Gauss-Bonnet theorem. By this way, we obtain an expression of
deflection angle and extend the study of deflection angle to asymptotically
non-flat black hole spacetimes. Some black holes as lens are considered such as
a static and spherically symmetric black hole in the conformal Weyl gravity and
a Schwarzschild-like black hole in bumblebee gravity. Our results are
consistent with the previous literature. In particular, we find that the
connection between Gaussian curvature and the radius of a circular orbit
greatly simplifies the calculation.
