Call one-half the length of the dark red line L. Since a tangent line is always perpendicular to the radius, this forms a right triangle such that

Therefore, the angle between the two radii is acos(r/R) and the area of the sector OAB is given by:

Furthermore: which means the area of triangle ABO is

The area of the pink area is the sector minus the triangle ABO, so you have:

(edited to clean up LaTeX)