# 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.[1]

## Formulas

On a unit sphere, the half-side formulas are[2]

where

*a*,*b*,*c*are the lengths of the sides respectively opposite angles*A*,*B*,*C*,- is half the sum of the angles, and

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

so that *a*,*b*,*c* all have length scales, instead of angular scales.

## References

- Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007),
*Handbook of Mathematics*, Springer, p. 165, ISBN 9783540721222 - Nelson, David (2008),
*The Penguin Dictionary of Mathematics*(4th ed.), Penguin UK, p. 529, ISBN 9780141920870.

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