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 • What "radius of circle" do you mean: the real ("planar") or the spherical one (measured along the surface)? – user Jan 18 at 12:47
• What exactly do you call a "circle segment on sphere" ? Your figure seems to mean a spherical cap, but this is unsure. – Yves Daoust Jan 18 at 16:34
• @user I have a spherical one (I measure it using Haversine formula) – Max Mikhalchuk Jan 18 at 16:49
• @YvesDaoust So, I have a map (Google Maps API) and I just draw a circle on it this way. And then I'm trying to find out the area of a circle segment – Max Mikhalchuk Jan 18 at 16:52
• @MaxMikhalchuk: I am not sure that the curvature of the Earth matters for this size. – Yves Daoust Jan 18 at 18:32

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 $$XYZ$$.