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

areacirclesgeometryspherical-geometry

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

enter image description here

Best Answer

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 $XYZ$.

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