Solve any triangle from 3 known values — supports SSS, SAS, ASA, AAS, and SSA (ambiguous case) with sides, angles, area, and visualization.
Solve any triangle instantly from any three known values. The Triangle Calculator handles all five solvable cases — SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), and the famously ambiguous SSA case which may yield zero, one, or two valid triangles. Get all six values (three sides, three angles), area via Heron formula, perimeter, inradius, circumradius, the three altitudes, plus a live SVG drawing scaled to fit. Switch freely between degrees and radians, and watch the figure update in real time as you type — no submit button required.
Any triangle has six parts — three sides (a, b, c) and three angles (A, B, C) opposite them. To solve a triangle uniquely you need exactly three of those values, and at least one must be a side. That gives five named cases: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a side not between them), and SSA (two sides and an angle not between them). The first four always produce exactly one triangle when the inputs are valid. SSA is the odd one out — it is the famous ambiguous case that can produce zero, one, or two triangles.
The law of cosines — c² = a² + b² − 2ab·cos(C) — is the workhorse for SSS and SAS. Rearranged, it gives any angle from three sides, or any side from two sides plus the included angle. The law of sines — a/sin(A) = b/sin(B) = c/sin(C) — handles ASA, AAS, and SSA. For area, Heron formula works once all three sides are known, while ½·ab·sin(C) is faster when you already have two sides and the included angle.
Every triangle has three altitudes (heights), one from each vertex perpendicular to the opposite side. They are all derived from the same area: hₐ = 2·Area/a, and similarly for h_b and h_c. The inradius r = Area/s is the radius of the largest circle that fits inside the triangle, tangent to all three sides — useful in tiling, fillet design, and structural engineering. The circumradius R = abc/(4·Area) is the radius of the unique circle passing through all three vertices.
Use the law of cosines when you have SSS or SAS — it directly gives you the missing side or angle from the three known values without ambiguity. Use the law of sines for ASA, AAS, and SSA cases where you have an angle and its opposite side. The calculator picks the right method automatically based on the mode you select.
SSA is the "ambiguous case" because two sides and a non-included angle can geometrically fit two different triangles — one acute and one obtuse on the unknown angle. When this happens, the calculator displays both solutions with their own diagrams. It can also produce zero solutions (when the side is too short) or exactly one (right-angle boundary or obtuse known angle).
Once all three sides are known, the calculator uses Heron formula: Area = √[s(s−a)(s−b)(s−c)] where s is the semi-perimeter. For SAS inputs it could equivalently use ½·a·b·sin(C); both methods give the same result, and Heron formula is numerically stable for all triangles.
The inradius (r) is the radius of the inscribed circle that fits inside the triangle, computed as Area/s. The circumradius (R) is the radius of the circumscribed circle passing through all three vertices, computed as abc/(4·Area). Both are useful in geometry proofs, engineering layouts, and CAD work.
The calculator labels the type automatically. A right triangle has one 90° angle, an obtuse triangle has one angle greater than 90°, and an acute triangle has all three angles below 90°. Combined with the side classification (equilateral, isosceles, or scalene), you get a complete description like "Acute Scalene" or "Right Isosceles."
Related: triangle calculator, solve triangle, law of cosines calculator, law of sines calculator, sss triangle, sas triangle, asa triangle, aas triangle, ssa ambiguous case, triangle area calculator, heron formula calculator, right triangle calculator, triangle angle calculator, triangle side calculator, trigonometry calculator