The solid angle of a region can be computed as the rotational transformation of an axis after its tip slews about the solid angle region once while its base remains fixed at the observer location. The transformation can be achieved by moving the tip of the axis from perimeter point to point sequentially using great circle arcs until it returns to its original orientation. This method works very well for spherical polygon shapes, but has reduced accuracy when the perimeter is curved. Increasing the number of points that define the perimeter improves accuracy but can also introduce numerical roundoff error. The proposed method uses both slewing and rotational motion of the axis to define the contour of the solid angle region more accurately. This new method results in greater accuracy while using fewer perimeter points. Numerical examples are included to illustrate the method.