Mollweide's formula

Figure 1 – A triangle. The angles α, β, and γ are respectively opposite the sides a, b, and c.

In trigonometry, Mollweide's formula, sometimes referred to in older texts as Mollweide's equations, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle. It can be used to check solutions of triangles.

Let a, b, and c be the lengths of the three sides of a triangle. Let α, β, and γ be the measures of the angles opposite those three sides respectively. Mollweide's formula states that

$\frac{a + b}{c} = \frac{\cos\left(\frac{\alpha - \beta}{2}\right)}{\sin\left(\frac{\gamma}{2}\right)}$

and

$\frac{a - b}{c} = \frac{\sin\left(\frac{\alpha - \beta}{2}\right)}{\cos\left(\frac{\gamma}{2}\right)}.$

Each of these identities uses all six parts of the triangle—the three angles and the lengths of the three sides.

See also

• Law of sines
• Law of cosines
• Law of tangents
• Law of cotangents

Additional reading

• H. Arthur De Kleine, "Proof Without Words: Mollweide's Equation", Mathematics Magazine, volume 61, number 5, page 281, December, 1988.

• Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 102
• Michael Sullivan, Trigonometry, Dellen Publishing Company, 1988, page 243.
• Ernest Julius Wilczynski, Plane Trigonometry and Applications, Allyn and Bacon, 1914, page 105