A curve traced in the plane by a point on a circle that rolls, without slipping, on a line. If the line is the x axis of a rectangular coordinate system, at whose origin O the moving point P touches the axis, parametric equations of the cycloid are x = a(θ − sin θ), y = a(1 − cos θ), when a is the radius of the rolling circle, and the parameter θ is the variable angle through which the circle rolls (see illus.). One arch is obtained where θ assumes all values from 0 to 2π. The length of an arch is 8a, and the area bounded by an arch and the x axis is 3πa² (three times the area of the rolling circle).