In this paper, a semi-Analytical framework, based on the scaled boundary finite element method (SBFEM), is proposed, to study interior acoustic problems in the mid-frequency range in two and three dimensions. The SBFEM shares the advantages of both the finite element method (FEM) and the boundary element method (BEM). Like the FEM, it does not require the fundamental solution (Green's function) and similar to the BEM only the boundary is discretized, thus reducing the spatial dimensionality by one. The solution within the domain is represented analytically, while on the boundary, it is represented by finite elements. Although different boundary representations are possible, only the Lagrangian description is investigated in this paper. The proposed framework is validated using closed-form solutions, and direct comparisons are made with the conventional FEM based on Lagrangian description; this will be demonstrated using two 2D cavities from literature as well as one 3D cavity. The improved accuracy and reduced computational time can be attributed to the semi-Analytical formulation combined with the boundary discretization. © 2021 Institute for Theoretical and Computational Acoustics, Inc.