Consider the unit sphere $S^2$ and a disc centred at $u\in S^2$. Also, let $A = \{x\in \mathbb{R}^3:x\cdot u = \cos r\}$ (where $r$ is the angle between $u$ and $v$), which is the plane whose nearest point to the origin is $(\cos r) u$. If $B = \{x \in \mathbb{R}^3: x\cdot u = 1\}$, which is the plane whose nearest point to the origin is $u$. Then the distance between A and B is $1 − \cos r$. Then the spherical disc $D(u,r)$ is equal to the portion of $S^2$ which lies between the parallel planes A and B, and so its area is $2\pi\Delta = 2π(1 − \cos r)$.
Why is the area of a spherical circle equal to $2\pi\Delta$? I don't seem to see how exactly this formula is justified.