Poisson's equation is very important in electrostatics, mechanical engineering and theoretical physics. In this paper, a novel semi-analytical mathematical method, namely scaled boundary finite-element method (SBFEM), is applied to solve Poisson's equation with Dirichlet and Neumann boundary conditions in bounded domain. The SBFEM weakens the governing differential equation in the circumferential direction and solves the weakened equation analytically in the radial direction, combining the advantages of the finite-element method and the boundary-element method. Three examples are calculated to demonstrate the excellent computation accuracy and efficiency of the present SBFEM approach, revealing the great potential of the SBFEM to solve more complex engineering problems.