How to use this calculator
- Enter or choose what you know.
- The triangles possible updates as you type. Use Copy result or Share to keep it.
Method
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
- 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
- Height h = b·sin A = 10 × 0.5 = 5. Since 5 < a = 6 < b = 10, two triangles exist.
- sin B = 10 × 0.5 ÷ 6 = 0.833333, so B₁ = 56.4427° and B₂ = 180° − 56.4427° = 123.5573°.
- Triangle 1: C = 180° − 30° − 56.4427° = 93.5573°, c = 6 × sin 93.5573° ÷ 0.5 ≈ 11.976879.
- Triangle 2: C = 26.4427°, c = 12 × sin 26.4427° ≈ 5.343629.
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).