Mollweide's formula

In trigonometry, Mollweide's formula, sometimes referred to in older texts as Molweide's equations, named after Karl Mollweide, is a relationship between sides and angles in a triangle.

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.

Mollweide's formula can be used to check solutions of triangles.

See also

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

References

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