In this thesis, we consider the Laplace quation on sector with various constant Dirichlet or Numann boundary conditions. Most of such problems have singularity in the solution. We first analyze the type of singularity on the corner and then survey some known methods to solve these problems. The boundary approximation method is used to compute some of their solutions with two singularities. Besides, a Laplace equation on a triangle with multiple solutions is solved by the method of separation of variables.