Based on the numerically solved Dirac equations, we study the electronic properties of the point vacancy of the graphene quantum dots with armchair boundary conditions under magnetic field. The size effect on the gap is analyzed. Without magnetic fields, quantum dot has finite energy gap which is proportional to the inverse of the radius of the dot. In the presence of the magnetic field, there appear Landau levels. The lowest Landau level has zero energy and is irrelevant to the magnetic field. With the increase of the magnetic field, the degeneracy of the Landau levels will increase. We further analyze the relationship between the lowest Landau level in the presence of magnetic field and the size of the quantum dot. The result shows that the degeneracy is linearly dependent on the magnetic field and the square of the radius. Our calculation will be possibly helpful in designing the device based on the graphene quantum dots.