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).


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

Thanks for help

enter image description here

  • $\begingroup$ What "radius of circle" do you mean: the real ("planar") or the spherical one (measured along the surface)? $\endgroup$ – user Jan 18 '19 at 12:47
  • $\begingroup$ What exactly do you call a "circle segment on sphere" ? Your figure seems to mean a spherical cap, but this is unsure. $\endgroup$ – Yves Daoust Jan 18 '19 at 16:34
  • $\begingroup$ @user I have a spherical one (I measure it using Haversine formula) $\endgroup$ – Max Mikhalchuk Jan 18 '19 at 16:49
  • $\begingroup$ @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 $\endgroup$ – Max Mikhalchuk Jan 18 '19 at 16:52
  • $\begingroup$ @MaxMikhalchuk: I am not sure that the curvature of the Earth matters for this size. $\endgroup$ – Yves Daoust Jan 18 '19 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 $XYZ$.


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