In this paper we give the new formulas for calculating the self-inductance for the circular coils of the rectangular cross sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the interval of the integration. From these new expressions we can obtain the special cases for the self-inductance of the thin disk pancake and the thin wall solenoid that confirm the validity of this approach. For the asymptotic cases, the new formula for the self-inductance of the thin wall solenoid is obtained for the first time in the literature. In this paper we do not use special functions such as the elliptical integrals of the first, second and third kind, Struve, and Bessel functions because that is very tedious work. The results of this work are compared with already different known methods and all results are in the excellent agreement. This is way we consider this approach as the novelty because of its simplicity in the self -inductance calculation of the previously mentioned configurations.