Lens-shaped intersection of two disks ($A=2r^2\arccos [d/(2r)]-(d/2)\sqrt{4r^2-d^2}$)

For two disks of equal radius $r$ whose centres are separated by $0\le d\le 2r$, their intersection consists of two circular segments. Each segment is a sector of angle $2\theta$ minus an isosceles triangle, where $\cos \theta =d/(2r)$. Hence its total area isThe limiting cases are $A=\pi r^2$ at $d=0$ and $A=0$ at $d=2r$. This geometric region is distinct from an optical lens.