The corner constant, which serves to correct the internal impedance per unit length of a conductor, is calculated in closed form for the limit of small ratios of internal to external magnetic permeabilities for a right angle corner. The perfectly conducting external tangential magnetic field distribution is used in an internal Green's theorem representation to determine the axial electric field. The corner constant is then determined by an integration of the surface axial electric field and the perfectly conducting magnetic field. Expansions of the axial electric field are also given because of its utility in future papers. A rotation of the radial coordinate into complex values is used to reduce the corner constant to a known complex phase times a real constant. This rotation is valid for general real ratios of internal to external magnetic permeabilities as a result of the analyticity of the fields in a sector above the positive real axis of the radial coordinate.