Area of a circle segment on sphere, given radius (meters) and central angle (degrees) Situation

I have a circle segment and some information about the circle it belongs to.

Given Information:


radius of the circle in meters
central angle in degrees
lat/long of all three points on the image below.


I need to calculate the area of the circle segment on sphere. It should depend on the radius of the Earth (like great-circle distance).

Problem

I have no idea how to do that and couldn't find any algorithms.

Thanks for help
Compute first the area of a full circle, then scale the result by $\theta/(2\pi)$ to get the area of the sector corresponding to central angle $\theta$. Finally, subtract from that the area of spherical triangle $XYZ$.
Compute first the area of a full circle, then scale the result by $\theta/(2\pi)$ to get the area of the sector corresponding to central angle $\theta$. Finally, subtract from that the area of spherical triangle $XYZ$.