[Math] Area of a Spherical Triangle from Side Lengths

geometryspherical-geometry

I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing to find the answer to the following embarrassingly easy (I think) question.

What is the area of a spherical triangle with all equal side lengths equal to $\pi / 3$?

I have tried using L'Juilier's Theorem, but the first term $$\tan(\frac{1}{2}s)=\tan(\frac{1}{2}(\frac{3\pi}{3}))=\tan(\frac{\pi}{2})$$
is undefined, where $s$ is the semiperimeter (perimeter divided by 2).

Any ideas for how to compute this?

Best Answer

From Mathworld (edited slightly):

Let a spherical triangle have angles $A$, $B$, and $C$ (measured in radians at the vertices along the surface of the sphere) and let the sphere on which the spherical triangle sits have radius $R$. Then the surface area $\Delta$ of the spherical triangle is $$\Delta=R^2(A+B+C-\pi).$$

Using the spherical Law of Cosines ($\cos c=\cos a\cos b+\sin a\sin b\cos C$, where $a$, $b$, and $c$ are the side lengths of the triangle), for your triangle, $A=B=C=\arccos\frac{1}{3}$.

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