Let and be the lengths of the three sides of a triangle.
Let and be the measures of the angles opposite those three sides respectively. Mollweide's formulas are
Because in a planar triangle these identities can alternately be written in a form in which they are more clearly a limiting case of Napier's analogies for spherical triangles (this was the form used by Von Oppel),
Dividing one by the other to eliminate results in the law of tangents,
In terms of half-angle tangents alone, Mollweide's formula can be written as
or equivalently
Multiplying the respective sides of these identities gives one half-angle tangent in terms of the three sides,
In spherical trigonometry, the law of cosines and derived identities such as Napier's analogies have precise duals swapping central angles measuring the sides and dihedral angles at the vertices. In the infinitesimal limit, the law of cosines for sides reduces to the planar law of cosines and two of Napier's analogies reduce to Mollweide's formulas above. But the law of cosines for angles degenerates to By dividing squared side length by the spherical excess we obtain a non-vanishing ratio, the spherical trigonometry relation:
In the infinitesimal limit, as the half-angle tangents of spherical sides reduce to lengths of planar sides, the half-angle tangent of spherical excess reduces to twice the area of a planar triangle, so on the plane this is:
and likewise for and
As corollaries (multiplying or dividing the above formula in terms of and ) we obtain two dual statements to Mollweide's formulas. The first expresses the area in terms of two sides and the included angle, and the other is the law of sines:
We can alternately express the second formula in a form closer to one of Mollweide's formulas (again the law of tangents):
A generalization of Mollweide's formula holds for a cyclic quadrilateral Denote the lengths of sides and and angle measures and If is the point of intersection of the diagonals, denote Then:[5]
Several variant formulas can be constructed by substituting based on the cyclic quadrilateral identities,
As rational relationships in terms of half-angle tangents of two adjacent angles, these formulas can be written:
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as approaches zero, a cyclic quadrilateral converges into a triangle and the formulas above simplify to the analogous triangle formulas. Relabeling to match the convention for triangles, in the limit and
^Sullivan, Michael (1988), Trigonometry, Dellen, p. 243
^Bradley, H. C.; Yamanouti, T.; Lovitt, W. V.; Archibald, R. C. (1921), "Discussions: Geometric Proofs of the Law of Tangents", American Mathematical Monthly, 28 (11–12): 440–443
^Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 105
De Kleine, H. Arthur (1988), "Proof Without Words: Mollweide's Equation", Mathematics Magazine, 61 (5): 281
Karjanto, Natanael (2011), "Mollweide's Formula in Teaching Trigonometry", Teaching Mathematics and Its Applications, 30: 70–74, arXiv:1808.08049, doi:10.1093/teamat/hrr008
Wu, Rex H. (2020), "Proof Without Words: The Mollweide Equations from the Law of Sines", Mathematics Magazine, 93 (5): 386, doi:10.1080/0025570X.2020.1817707