What "Solving a Triangle" Means
A triangle has six parts — three sides, three angles — and knowing the right three determines the rest. This solver accepts any valid combination:
| Given | Name | Solved with |
|---|---|---|
| Three sides | SSS | Law of cosines |
| Two sides + included angle | SAS | Law of cosines, then sines |
| Two angles + included side | ASA | Angle sum, law of sines |
| Two angles + non-included side | AAS | Angle sum, law of sines |
| Two sides + non-included angle | SSA | Law of sines — the ambiguous case (0, 1 or 2 solutions) |
Three angles alone (AAA) fix only the shape, not the size — no unique triangle exists, and the solver will tell you so.
The Two Laws Doing the Work
Law of cosines: c² = a² + b² − 2ab·cos C
Cosines generalizes Pythagoras (set C = 90° and the last term vanishes) and handles the cases sines can't start; sines finishes efficiently once a side-angle pair is known. Angles always total 180°, which supplies each third angle for free.
Worked Example: SAS
Given b = 8, c = 5, included angle A = 60°:
| Find a | a² = 64 + 25 − 2·8·5·cos 60° = 89 − 40 = 49 | a = 7 |
| Find B | sin B = b·sin A / a = 8×0.866/7 | B ≈ 81.8° |
| Find C | 180 − 60 − 81.8 | C ≈ 38.2° |
| Area | ½·b·c·sin A = ½·8·5·0.866 | ≈ 17.32 |
| Perimeter | 7 + 8 + 5 | 20 |
Every solve in the tool shows this style of breakdown — usable directly as homework working.
Area, Three Ways
Two sides + angle: A = ½·a·b·sin C
Heron (three sides): s = (a+b+c)/2; A = √(s(s−a)(s−b)(s−c))
The solver picks whichever your inputs allow. Heron's is the SSS workhorse: sides 7-8-5 → s = 10 → √(10·3·2·5) = √300 ≈ 17.32 — matching the SAS result above, as it must. For everyday base-and-height work across eight shapes — rectangles, circles, trapezoids and more — the Area Calculator is the tool built for it.
Validity and the Ambiguous Case
- Triangle inequality: each side must be shorter than the other two combined — 3, 4, 9 is impossible and the solver says so rather than emitting nonsense.
- SSA ambiguity: two sides and a non-included angle can match two different triangles (or one, or none). The solver detects the case and reports both solutions when they exist — the classic exam trap, automated away.
- Degenerate flatness: sides like 2, 3, 5 collapse to a line (area zero) — flagged, not solved.
Right triangles, as the special case, also yield to Pythagoras directly — and quadratic setups from geometry land in the Quadratic Solver's lap.
The Bottom Line
Two laws and the 180° angle sum solve every solvable triangle — the craft is knowing which law starts from your given pattern, and the solver above automates exactly that, validity checks and ambiguous cases included. Enter what you know; get the five missing pieces with the working shown. And when a flat shape gains depth, the Volume Calculator picks up from there.