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Fig 1a shows two planes at right angles and centered at the origin. In 2b, the planes intersect with a sphere, also centered at the origin. The intersection of each plane with the sphere forms a great circle. So each great circle lies in its own plane (1c). The angle between two great circles is measured as the angle between their planes (1d). Opposite angles between intersecting planes are equal. So in fig 1d, a = a' and b = b'.


Fig 1f. Two great circles will always have a pair of poles in common, 1 diameter apart, and will divide the sphere into four Lunes. Thus point B is “diametrically opposite” to point A.

The area of a lune depends on the angle between its great circles. Since opposite angles between intersecting planes are equal, each white lune has the same area, and each blue lune has the same area. 1 white lune + 1 blue lune = 1 hemisphere.

Recall that the area A of a sphere is: A = 4piR².

If the angle between two great circles is 90° (or pi over -two radians), then the area of the lune will be one quarter of the entire sphere, or piR².

For any angle theta, the area A of a lune is:

A = 2R² theta,   (if theta is in radians)
A = pitheta° / 90°   (if theta is in degrees)

[Ref: ‘The VNR Concise Encyclopedia of Mathematics’ p.262. W. Gellert, H. Küstner, M. Hellwich, H.Kästner (K.A Hirsch, H. Reichart), Bibliographisches, Institut Leipzig 1975]