Quick answer
Pick the calculation mode that matches what you know — three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA), two angles and a non-included side (AAS), or two sides and a non-included angle (SSA). The calculator applies the Law of Cosines or Law of Sines to solve for every remaining side, angle, the area, and the perimeter.
Key takeaways
- Any triangle needs exactly three known values (in a valid combination) to be fully solved.
- The Law of Cosines connects three sides and one angle; the Law of Sines connects sides and angles proportionally to their opposite angle's sine.
- The three interior angles always sum to exactly 180°, no matter the triangle's shape or size.
- SSA input is the one case that can be ambiguous — it may describe zero, one, or two valid triangles.
- Heron's formula finds area directly from three sides, without needing any angle.
Choosing a method: SSS, SAS, ASA, AAS, SSA
Each mode describes a different combination of known sides and angles. Match what you have to a row below:
| Mode | You know | Solved with |
|---|---|---|
| SSS | All three sides | Law of Cosines (angles), Heron's formula (area) |
| SAS | Two sides + the angle between them | Law of Cosines (third side), then Law of Cosines again for angles |
| ASA | Two angles + the side between them | Triangle sum theorem (third angle), Law of Sines (sides) |
| AAS | Two angles + a side not between them | Triangle sum theorem (third angle), Law of Sines (sides) |
| SSA | Two sides + a non-included angle | Law of Sines — may have 0, 1, or 2 solutions |
Law of Sines and Law of Cosines
The Law of Cosines generalizes the Pythagorean theorem to any triangle:
a² = b² + c² − 2bc·cos(A)
The Law of Sines relates each side to the sine of its opposite angle:
a / sin(A) = b / sin(B) = c / sin(C)
Worked example (ASA): Angle A = 40°, side c = 10, Angle B = 60°
- Angle C = 180° − 40° − 60° = 80°
- Side a = c × sin(A) / sin(C) = 10 × sin(40°) / sin(80°) ≈ 6.53
- Side b = c × sin(B) / sin(C) = 10 × sin(60°) / sin(80°) ≈ 8.79
- Area = 0.5 × a × b × sin(C) ≈ 0.5 × 6.53 × 8.79 × sin(80°) ≈ 28.27
The ambiguous SSA case
When you know two sides and a non-included angle, the height h = b × sin(A) determines how many triangles fit those measurements. Example: side b = 10, Angle A = 30° (so h = 10 × sin(30°) = 5):
| Side a | Comparison to h = 5 | Result |
|---|---|---|
| 4 | a < h | No triangle exists |
| 5 | a = h | Exactly one triangle (a right triangle) |
| 7 | h < a < b | Two possible triangles (ambiguous) |
| 12 | a ≥ b | Exactly one triangle |
This calculator returns the valid solution set for your inputs; when two triangles are mathematically possible, it reports the acute-angle solution.
Common mistakes
- Entering three sides that violate the triangle inequality — the two shorter sides must add up to more than the longest side, or no triangle can form.
- Mixing up SAS and ASA — SAS needs the angle sandwiched between the two given sides, while ASA needs the side sandwiched between the two given angles.
- Forgetting that SSA can be ambiguous and assuming there is only one possible triangle.
- Working in the wrong angle unit — this calculator expects degrees, not radians.
Related calculators
- Right Triangle Calculator — for triangles with a known 90° angle
- Pythagorean Theorem Calculator — for finding a missing side of a right triangle
- Area Calculator — for area of triangles and other shapes
- Distance Calculator — for the distance between two coordinate points