We detailedly describe a recently-developed semi-analytical method to quantitatively calculate light transmission properties of hollow-core anti-resonant fibers (HC-ARFs). Formation of equiphase interface at fiber's outermost boundary and outward light emission ruled by Helmholtz equation in fiber's transverse plane constitute the basis of this method. Our semi-analytical calculation results agree well with those of precise simulations and clarify the light leakage dependences on azimuthal angle, geometrical shape and polarization. Using this method, we show investigations on HC-ARFs having various core shapes (e.g., polygon, hypocycloid) with single-and multi-layered core-surrounds. The polarization properties of ARFs are also studied. Our semi-analytical method provides clear physical insights into the light guidance in ARF and can play as a fast and useful design aid for better ARFs.