# Law of Sines Calculator

> Solve a triangle with the law of sines from two sides and a non-included angle (finding 0, 1 or 2 triangles) or from two angles and a side.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator

The law of sines says a ÷ sin A = b ÷ sin B = c ÷ sin C. Given two sides and an angle that is not between them, there may be no triangle, one, or two, and this calculator finds them all.

Also called: Sine rule calculator, SSA triangle calculator.

## Inputs
- What you know (options: Two sides + angle (SSA), Two angles + side (AAS); default ssa)
- Angle A (unit °; default 30)
- Side a (opposite angle A) (default 6)
- Side b (default 10)
- Angle A (unit °; default 30)
- Angle B (unit °; default 45)
- Side a (opposite angle A) (default 10)

## Results
- Triangles possible
- Number of triangles
- Angle B (first triangle)
- Angle C (first triangle)
- Side b
- Side c (first triangle)
- Second triangle
- a ÷ sin A (= 2R)

## Formula

```
a ÷ sin A = b ÷ sin B = c ÷ sin C = 2R;  SSA: sin B = b·sin A ÷ a, and B may be B₁ or 180° − B₁;  C = 180° − A − B
```

Where:
- a, b, c: sides opposite angles A, B, C
- R: the circumradius
- h = b·sin A: the height that decides how many SSA triangles exist

## Worked example
1. Height h = b·sin A = 10 × 0.5 = 5. Since 5 < a = 6 < b = 10, two triangles exist.
2. sin B = 10 × 0.5 ÷ 6 = 0.833333, so B₁ = 56.4427° and B₂ = 180° − 56.4427° = 123.5573°.
3. Triangle 1: C = 180° − 30° − 56.4427° = 93.5573°, c = 6 × sin 93.5573° ÷ 0.5 ≈ 11.976879.
4. Triangle 2: C = 26.4427°, c = 12 × sin 26.4427° ≈ 5.343629.

**Result:** With A = 30°, a = 6 and b = 10 there are two triangles: c ≈ 11.977 (B ≈ 56.44°) or c ≈ 5.344 (B ≈ 123.56°).

## What the result means

In the SSA case compare side a with the height h = b·sin A: a < h gives no triangle, a = h one right triangle, h < a < b two triangles, and a ≥ b one triangle. If a drawing or context tells you whether B is acute or obtuse, choose the matching triangle.

## Assumptions and limits
- Angles are in degrees.
- Angle A is opposite side a; in SSA mode angle A is not between the two given sides.
- When a is extremely close to the height h, rounding in the inputs can switch the answer between one and two triangles.

## Questions

### When should I use the law of cosines instead?

Use the law of cosines when you know all three sides (SSS) or two sides and the angle between them (SAS). The Triangle Calculator handles those cases.

### Why can SSA give two triangles?

Side a can swing to meet the base line at two points when it is longer than the height but shorter than side b. Both positions satisfy the same sine value, because sin B = sin(180° − B).

## Related calculators
- [Triangle Calculator](https://calculators.geekmidst.com/calculators/triangle-calculator)
- [Triangle Circumradius Calculator](https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator)
- [Right Triangle Calculator](https://calculators.geekmidst.com/calculators/right-triangle-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator