Numerical simulations of body-of-revolution geometries for scattering and radiation problems are presented. The formulation consists of a finite element-boundary integral (FE-BI) method which is based on a finite element method that uses higher order nodal-based scalar basis functions for the azimuthal field component and higher order edge-based vector basis functions for the transverse field. This formulation, when combined with a symmetric FE-BI hybridization scheme, yields a final system of equations that is more accurate than earlier first-order formulations. Numerical examples are given to demonstrate the accuracy and capabilities of the higher order solution