Areas of the Lens and Two Lunes of Two Intersecting Circles

Intersecting two overlapping circles determines three distinct sets: the gray lens and the two lunes, in green and purple. Their areas are computed directly (without using calculus) in terms of the areas of the circular sectors and segments determined by the white, yellow, and black lines in the figure. As you vary the parameters, you can see the corresponding changes in the numerical values of the three areas.