In this talk I will describe a symplectic approach to K¨ahler geometry on toric orbifolds, based on the existence of action-angle coordinates. The main result of this approach is an effective parametrization of all invariant K¨ahler metrics on a symplectic toric orbifold, via smooth functions on the corresponding moment polytope. As an application I will give a simple explicit description of recent work of R. Bryant, producing interesting new families of extremal K¨ahler metrics. In particular, one obtains an extremal K¨ahler metric on any weighted projective space.