Mollweide's Formula
Mollweide's formula relates sides and angles of a triangle in a single equation.
Useful for checking triangle solutions with examples.
The Formula
Mollweide's formula (also called Mollweide's equation) provides a relationship involving all three sides and all three angles of a triangle in a single equation. It was published by German mathematician Karl Mollweide in 1808.
There is also a companion form: (a + b) / c = cos((A - B) / 2) / sin(C / 2). Together, these two equations are sometimes called Mollweide's pair.
The primary use of Mollweide's formula is as a check on solutions obtained by other methods. After solving a triangle using the law of sines or law of cosines, you can substitute all six values (three sides and three angles) into Mollweide's equation. If both sides are equal, your solution is correct. If they differ, there is an error somewhere in your calculations. This makes it an invaluable verification tool for navigators and surveyors.
Variables
| Symbol | Meaning |
|---|---|
| a, b, c | Sides of the triangle opposite to angles A, B, C respectively |
| A, B, C | Angles of the triangle (A + B + C = 180°) |
Example 1
Verify the triangle with a = 8, b = 5, c = 9, A = 62.2°, B = 33.6°, C = 84.2°.
Left side: (a - b)/c = (8 - 5)/9 = 3/9 = 0.3333
Right side: sin((A-B)/2) / cos(C/2) = sin((62.2-33.6)/2) / cos(84.2/2)
= sin(14.3°) / cos(42.1°) = 0.2470 / 0.7420
= 0.3329
Left ≈ 0.333, Right ≈ 0.333. The values match, confirming the triangle is correct.
Example 2
Using the companion formula, verify the same triangle: (a + b)/c = cos((A-B)/2) / sin(C/2).
Left side: (a + b)/c = (8 + 5)/9 = 13/9 = 1.4444
Right side: cos((62.2-33.6)/2) / sin(84.2/2) = cos(14.3°) / sin(42.1°)
= 0.9690 / 0.6704
= 1.4454
Left ≈ 1.444, Right ≈ 1.445. Close match confirms the solution.
When to Use It
Mollweide's formula is primarily used for verification and error checking.
- Checking solutions after solving a triangle with the law of sines or cosines
- Surveying and navigation — verifying field measurements
- Teaching trigonometry — demonstrating relationships between sides and angles
- Detecting computational errors in triangle calculations