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]


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


  • 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.

See also


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