Dividing a circle into areas "F" can also be derived, which will solve the problem. Its vertices include the "n" points on the circle, referred to as the exterior vertices, as well as the interior vertices, the intersections of distinct chords in the interior of the circle. The "generic intersection" assumption made above guarantees that each interior vertex is the intersection of no more than two chords. Thus the main task in determining "V" is finding the number of interior vertices. As a consequence of the lemma, any two intersecting chords will uniquely determine an interior vertex. These chords are in turn uniquely determined by
Area of a circle version of the divergence theorem), in a way that avoids all mention of trigonometry and the constant . Consider the vector field formula_27 in the plane. So the divergence of r is equal to two, and hence the area of a disc "D" is equal to By Green's theorem, this is the same as the outward flux of r across the circle bounding "D": where n is the unit normal and "ds" is the arc length measure. For a circle of radius "R" centered at the origin, we have formula_30 and formula_31, so the above equality is The integral of