Half-side formula
[Image:Law-of-haversines.svg|right|thumb|Spherical triangle]
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.
For a triangle on a sphere, the half-side formula is
where are the angular lengths of the sides opposite angles respectively, and is half the sum of the angles. Two more formulas can be obtained for and by permuting the labels
The polar dual relationship for a spherical triangle is the half-angle formula,
where semiperimeter is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels
Half-tangent variant
The same relationships can be written as rational equations of half-tangents. If and then the half-side formula is equivalent to:and the half-angle formula is equivalent to: