Choosing the right known set
Any triangle can be solved once you know three of its six parts, provided at least one of them is a side. This solver covers the four configurations that come up most often, named by the order their known parts go around the triangle.
Match your data to a mode before entering numbers, because each mode reads a different group of fields and ignores the rest.
- SSS — all three sides a, b and c.
- SAS — two sides a and b with the included angle C between them.
- ASA — two angles A and B with the side c that lies between them.
- AAS — two angles A and B with side a, which is opposite one of those angles.
The laws of sines and cosines
When an angle sits between two known sides (SAS) or all three sides are known (SSS), the law of cosines, c² = a² + b² − 2·a·b·cos(C), supplies the first missing piece.
When you start from two angles (ASA or AAS), the law of sines takes over: a/sin A = b/sin B = c/sin C. The third angle is simply 180° minus the other two, and the remaining sides scale in proportion to the sine of their opposite angles.
Why two angles must total under 180°
The three interior angles of any triangle add up to exactly 180°. So in the angle-based modes, the two angles you supply must sum to less than 180°, leaving a positive amount for the third.
If you enter angles that meet or exceed 180° together, no valid triangle exists and the solver reports an error rather than a nonsensical result.
Reading the output
The solver returns all three sides, all three angles in degrees, and the area. The largest angle always faces the longest side, a handy check that the numbers are consistent.
This tool assumes a single, well-defined triangle. The ambiguous SSA case — two sides and a non-included angle, which can yield two different triangles — is deliberately not offered here.
Formula
Law of sines: a/sin A = b/sin B = c/sin C. Law of cosines: c² = a² + b² − 2·a·b·cos C.Frequently asked questions
- Which fields does each mode use?
- SSS: a, b, c. SAS: a, b, angle C. ASA: angle A, angle B, side c. AAS: angle A, angle B, side a. Other fields are ignored.
- Why must two angles be under 180° together?
- The three interior angles of a triangle sum to 180°, so any two given angles must add up to less than that.