Triangle Solver
Solve any triangle from any three known values — sides, angles, area, perimeter or height, including the ambiguous SSA case. Returns all sides and angles, area, perimeter, and the inscribed and circumscribed circles.
Solving a Triangle
A triangle has six parts: three sides a, b, c and the three angles A, B, C opposite them. Knowing any three parts — as long as at least one is a side — usually fixes the other three.
The SSA case is special: two sides and an angle that is not between them can give no triangle, exactly one, or two different triangles. This calculator shows both when that happens.
Live Diagram
to scaleWhich Law to Use
| Case | Given | Strategy |
|---|---|---|
| SSS | 3 sides | cosines for each angle |
| SAS | 2 sides + angle between | cosines for 3rd side, then angles |
| ASA | 2 angles + side between | angle sum, then sines |
| AAS | 2 angles + other side | angle sum, then sines |
| SSA | 2 sides + other angle | sines — 0, 1 or 2 solutions |
Key Formulas
Pythagoras for any triangle: when C = 90° the last term vanishes. Use it when you know all three sides, or two sides and the angle between them.
Each side divided by the sine of its opposite angle gives the same number — the diameter of the circumscribed circle.
With s = ½(a+b+c) the semi-perimeter, Heron’s formula needs only the three sides.
Sources & Verification
- J. Stewart, L. Redlin & S. Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage, 2016 — Sec. 6.2 (Trigonometry of Right Triangles), Sec. 6.5 (The Law of Sines) and Sec. 6.6 (The Law of Cosines).