# Half-side formula

In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles.

## Formulas

On a unit sphere, the half-side formulas are

{\begin{aligned}\tan \left({\frac {a}{2}}\right)&=R\cos(S-A)\\[8pt]\tan \left({\frac {b}{2}}\right)&=R\cos(S-B)\\[8pt]\tan \left({\frac {c}{2}}\right)&=R\cos(S-C)\end{aligned}} where

• a, b, c are the lengths of the sides respectively opposite angles A, B, C,
• $S={\frac {1}{2}}(A+B+C)$ is half the sum of the angles, and
• $R={\sqrt {\frac {-\cos S}{\cos(S-A)\cos(S-B)\cos(S-C)}}}.$ The three formulas are really the same formula, with the names of the variables permuted.

To generalize to a sphere of arbitrary radius r, the lengths a,b,c must be replaced with

• $a\rightarrow a/r$ • $b\rightarrow b/r$ • $c\rightarrow c/r$ so that a,b,c all have length scales, instead of angular scales.

## See also

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