This study examines the problem of the intersection of a spiral curve with a circle, which especially occurs in the design of junctions. To achieve the requirements, an iterative method which has been programmed, is suggested with full mathematical formulas. Assessments of selected examples are made, and the results are tabulated. The intersection points are derived by starting first from a upper limit and second from a lower limit. If the same points are obtained when the iteration starts from both the upper and lower limits, the circle intersects the spiral at one point. If they are different points, the circle intersects the spiral at two points. However, if the iteration does not converge there are two cases for the problem: the circle either is tangent to the spiral or does not intersect it. In the tangent case, a method is suggested in order to find the tangent points at a sufficient precision.