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Consider a spherical triangle with vertices $A, B$ and $C$, respectively. How to determine its area?

I know the formula:

$A = E R^2$,

where $R$ is radius of sphere, and $E$ is the excess angle of $(a + b + c - \pi)$, but how to determine the angles between $ABC, ACB$ and $BAC$?

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How are A, B, C given? As vectors in three-space? –  Sam Nead Nov 11 '10 at 13:51

3 Answers 3

If you know the angular distance between the points, L'Huilier's Formula (cited in Derek Jennings' answer) gives $$ \tan\left(\frac{E}{4}\right)=\sqrt{\tan\left(\frac{s}{2}\right)\tan\left(\frac{s-a}{2}\right)\tan\left(\frac{s-b}{2}\right)\tan\left(\frac{s-c}{2}\right)}\tag{1} $$ where $a=\operatorname{ang}(B,C)$, $b=\operatorname{ang}(C,A)$, $c=\operatorname{ang}(A,B)$, and $s=\frac{a+b+c}{2}$.

If $A$, $B$, and $C$ are given as points in $\mathbb{R}^3$, then you can use $(1)$ with $$ \operatorname{ang}(A,B)=\cos^{-1}\left(\frac{A\cdot B}{|A||B|}\right)\tag{2} $$ or, to answer the question you asked, you can compute $$ \angle CAB=\cos^{-1}\left(\frac{(C\;A\cdot A-C\cdot A\;A)\cdot(B\;A\cdot A-B\cdot A\;A)}{|C\;A\cdot A-C\cdot A\;A||B\;A\cdot A-B\cdot A\;A|}\right)\tag{3} $$ and simply compute $E=\angle CAB + \angle ABC + \angle BCA - \pi$.

If $A$, $B$, and $C$ are given in other formats, there are probably ways to handle those, too, but more specific information would be needed.

Addition: Using cross products simplifies $(3)$ a bit: $$ \angle CAB=\cos^{-1}\left(\frac{(C\times A)\cdot(B\times A)}{|C\times A||B\times A|}\right)\tag{4} $$

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The angle between two great circles is equal to the angle between the planes containing them. Such a plane always passes through the centre of the sphere, say $O$. So, for example, angle $a$ is the angle between two planes, one of which passes through $A$, $B$, and $O$, and the other passes through $A$, $C$, and $O$.

The angle between two planes, in turn, equals the angle between their normals, and you can determine the normal from the three points by using the cross product.

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The area of a spherical triangle can also be calculated using the lengths of its sides, as in this Dr.Math link.

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