Spherical projection of a tangent (on the triangle center) right isosceles triangle.

Question

I am trying to calculate the visibility of a surface by an observer. To do this I discretized the surface in isosceles right triangles. (half of a square). I need to calculate the projection of these triangles onto the sphere which has the observer as its center and which passes through the center of the triangle. I think that the projection on the tangent plane is elementary and therefore we can also assume that we have a triangle located on the plane tangent to the sphere and with the center on it (but I don't think the properties of the triangle are maintained in this case). What interests me is exclusively the area of ​​the projection (of the spherical triangle) as a function of the radius of the sphere and of the angle (smaller) between the plane of the surface and the directional between the center of the triangle and that of the sphere, and of any other information about the triangle (I have all his vertices). My knowledge does not allow me to find the solution! Can anyone help me out?