In the current study, the Laplace equation is solved for rectangular and elliptical computational domains by using the boundary element method (BEM). For this accomplishment, 120 different aspect ratios in a rectangular and elliptical computational domains are designed. The Dirichlet and Neumann boundary conditions are used respectively for a rectangular and elliptical domains. Also, the Gaussian quadrature integral method is applied to solve the influence coefficient matrix in BEM. To assess a different aspect ratio on the potential solution, two different measurement positions are intended. According to our finding, with an increase of the aspect ratio, the potential value is increased for both rectangular and elliptical domains. However, a potential increment with aspect ratio enhancement is more visible in the elliptical domain.