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 a, b, c are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles A, B, C respectively, and is half the sum of the angles. Two more formulas can be obtained for b and c by permuting the labels A, B, C. 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 A, B, C.

Half-tangent variant

The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If and then the half-side formula is equivalent to: and the half-angle formula is equivalent to: