Library / Foundations: Maths / FMA-14
Foundations: Maths
Sine rule
Side of any triangle from another side and two angles.
- Ref
- FMA-14
- Rev
- v1
- Field
- Foundations: Maths
- Result
- b [m]
Match each side to its opposite angle. The calculator finds side b from side a and angles A and B (both below 90° here).
In any triangle, each side divided by the sine of its opposite angle gives the same number.
Calculator
- Angle C (if A + B < 180°)80 °
- Common ratio a / sin A10.89 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Use it when you know two angles and a side, or two sides and a non-included angle.
- Triangulation of a target from two bearings is a sine-rule problem.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Side a | 1 to 100 m | 1.347 | 134.7 | 133.4 | +10% | +10 % |
| A Angle A | 20 to 80 ° | 6.156 | 17.72 | 11.57 | +10% | −7.47 % |
| B Angle B | 20 to 80 ° | 3.725 | 10.72 | 7 | +10% | +5.49 % |
Sweep of Angle B
| Angle B (°) | Side b (m) |
|---|---|
| 20 | 3.725 |
| 27.5 | 5.028 |
| 35 | 6.246 |
| 42.5 | 7.357 |
| 50 | 8.342 |
| 57.5 | 9.185 |
| 65 | 9.87 |
| 72.5 | 10.39 |
| 80 | 10.72 |
Sweep one variable
Variables and how each one contributes
Side b
What the equation gives you. Everything else feeds it.
Side a
Side opposite angle A. Directly proportional.
Angle A
Angle opposite side a. Larger A shrinks b.
Angle B
Angle opposite side b. Larger B grows b.
Worked example with the values above
- Common ratio: a / sin A = 7 / 0.6428 = 10.89.
- b = ratio × sin B = 10.89 × 0.866.
- b = 9.431 m.
Assumptions and limits
- Plane triangle.
- Angle and side correctly paired (opposite).
Common mistakes
- Pairing a side with the wrong angle.
- Ambiguous case: two triangles can fit two sides and a non-included angle.
Statistics and uncertainty
Errors in the angles feed through the sine ratio; for small A the denominator shrinks, so baseline errors become large in the computed range.
Where it is used
- Surveying and triangulation.
- Passive location from two bearings.
Maths you need
- Trigonometry: sine of an angle
- Triangles: angles sum to 180°