Fig 3a shows a spherical triangle I with angles abc. This triangle is part of the lune with poles R and R' (3b). This lune consists of spherical triangles I + IV (3c). At pole R, we can see the lune has angle c. Therefore the area of the lune with poles R and R' = 2R²c.
I + IV = 2R²c.
By the same method, the lune with poles Q and Q' consists of spherical triangles I + II. So the area of this lune is:
I + II = 2R²b.
And the area of the lune with poles P and P' is:
I + III' = 2R²a.
Note that in 3c, each spherical triangle at the front I, II, III, IV, has a diametrically opposite triangle at the back I', II', III', IV'. Thus III and III' are congruent, which means the last equation can be rewritten as:
I + III = 2R²a.
Adding the 3 equations together gives:
3 • I + II + III + IV = 2R²(a + b + c)
It is obvious that I + II + III + IV is a hemisphere:
I + II + III + IV = 2R²
And by symmetry about the center, I + II + III + IV' is also a hemisphere because IV' is congruent and diametrically opposed to IV.
I + II + III + IV = I + II + III + IV' = 2R²
Note: this applies this to any congruent pair of spherical triangles. For example, in the case of II and II', we can say:
I + II + III + IV = I + II' + III + IV = 2R²
2 • I + 2R² = 2R²(a + b + c)
I = R²(a + b + c - ) = (R² / 180°)(a° + b° + c° - 180°)
Therefore, the area of I is:
I = R² [ (a + b + c) - ]
The area A of a spherical triangle is:
A = R² [ (a + b + c) - ] (if a, b and c are in radians).
A = R² [ (a° + b° + c°) - 180° ] (if a, b and c are in degrees).
It has already been shown that the sum of the angles of a spherical triangle are together greater than two right angles (). The difference between this sum and (or 180°) is called the spherical excess.
The area of a spherical triangle is equal to its spherical excess.
[Ref: ‘The VNR Concise Encyclopedia of Mathematics’ p.263. W. Gellert, H. Küstner, M. Hellwich, H.Kästner (K.A Hirsch, H. Reichart), Bibliographisches, Institut Leipzig 1975]
[Ref: ‘Geometry’ p.341-342. D. A. Brannan, M. F. Esplen, J. J. Gray, Cambridge University Press]