This paper deals with a path planning problem for turning motion of a car-like vehicle. We propose a turning method which finds a curvature continuous optimal path between two positions for a car-like vehicle. This method suggests two kinds of turning case; 1) Turning path with an arc of minimum turning radius ; 2) Turning path without an arc of minimum turning radius. In each case, we derive an closed form functions, an object function and a set of related constraints. By using those equations, optimal turning path candidates corresponding to each case are made. Furthermore, we find a final distance optimal turning path among above candidates for each case. Our algorithm is novel and more challenging compared to existing path planning algorithms in the sense that it deals more complicated problem, a turning motion. Moreover, we expect that the proposed method is helpful in various ways such as an improvement of existing path planners, like sampling based planners.