L'Huilier's Theorem

Let a Spherical Triangle have sides of length $a$, $b$, and $c$, and Semiperimeter $s$. Then the Spherical Excess $\Delta$ is given by

$$\tan({\textstyle{1\over 4}}\Delta)=\sqrt{\tan({\textstyle{1\... ...\textstyle{1\over 2}}(s-b)]\tan[{\textstyle{1\over 2}}(s-c)]}.$$

See also Girard's Spherical Excess Formula, Spherical Excess,Spherical Triangle

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 148, 1987.