Motivated by the development of photonic band gap materials, we consider the scattering and computation of resonances by a thin structure heterogeneous structure – typically periodic with a defect. Helmholtz’s equation with variable coefficient models the wave phenomena. The scatterer is assumed to have a high index of refraction while at the same time it is very small in one of the dimensions. We show that if the index scales as O(1/h), where h is the thickness of the scatterer, then an approximate solution, based on perturbation analysis, can be obtained. The approximate solution consists of a leading order term plus a corrector, each of which solves an integral equation in 2D for a 3D problem. We provide error analysis on the approximation and compare the computation of the resonances by two proposed approaches.