Triangle Solver

"Solving a triangle" means: given three of the six parts of a triangle (three sides + three angles), find the other three. Whether your three knowns are SSS / SAS / ASA / AAS / SSA, the Triangle Solver picks the right formula automatically. This tutorial walks through what happens behind each click, so you know which inputs produce a unique triangle and which produce zero or two solutions.

The six parts of a triangle

Every triangle has three sides (labelled a, b, c) and three angles (A, B, C) — each angle opposite the side of the same letter. Solving a triangle requires at least three known parts, with at least one being a side (because three angles without any side define infinitely many similar triangles).

The two master formulas

Every method reduces to one of two relationships:

• Law of Cosines: c² = a² + b² − 2ab·cos(C). Solves for a side when you have two sides and the included angle, or solves for an angle when you have all three sides.
• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Solves for a side when you have a side, its opposite angle, and one more angle.

SSS — three sides given

Inputs: a, b, c. Outputs: A, B, C, area, perimeter.
The solver computes cos(C) = (a² + b² − c²) / (2ab), takes arccos to get C, repeats for A or B, then uses A + B + C = 180° for the last. Heron's Formula gives area from just the three sides — no height needed.

Example: a = 5, b = 7, c = 9. cos(C) = (25 + 49 − 81) / 70 = −0.1 → C ≈ 95.74°. sin(A) / 5 = sin(95.74°) / 9 → A ≈ 33.56°. B = 180° − 95.74° − 33.56° = 50.70°. Area = √(s(s−a)(s−b)(s−c)) where s = 10.5 → Area ≈ 17.41.

SSS always gives a unique triangle provided the triangle inequality holds (each side < sum of the other two).

SAS — two sides + included angle

Inputs: two sides and the angle between them (e.g. a, b, C).
Law of Cosines gives the third side: c² = a² + b² − 2ab·cos(C). Then Law of Sines gives one of the other angles, and the third is 180° minus the sum.

Example: a = 8, b = 10, C = 60°. c² = 64 + 100 − 160·cos(60°) = 84 → c ≈ 9.17. sin(A) / 8 = sin(60°) / 9.17 → A ≈ 49.11°. B = 70.89°.

ASA — two angles + included side

Inputs: two angles + the side between them (e.g. A, B, c).
Third angle = 180° − A − B. Then Law of Sines for each remaining side.

Example: A = 50°, B = 60°, c = 12. C = 70°. a = 12 × sin(50°) / sin(70°) ≈ 9.78. b ≈ 11.06.

AAS — two angles + a non-included side

Inputs: two angles and a side opposite one of them (e.g. A, B, a). Same as ASA: compute third angle, then Law of Sines.

SSA — the ambiguous case

Inputs: two sides + an angle opposite one of them (e.g. a, b, A — but the angle isn't between the two sides).
This is the only case that can produce zero, one, or two valid triangles. The solver checks sin(B) = b × sin(A) / a:

• If sin(B) > 1 → no triangle exists (side b too long for angle A).
• If sin(B) = 1 → one right triangle (B = 90°).
• If sin(B) < 1 → two candidates B₁ = arcsin(...) and B₂ = 180° − B₁. Both are valid if A + B<180° in each case.

Example with two solutions: a = 6, b = 8, A = 35°. sin(B) ≈ 0.7648. B₁ ≈ 49.86° (acute), B₂ ≈ 130.14° (obtuse). A + B₁ = 84.86° and A + B₂ = 165.14° — both < 180°, so both are valid triangles. The solver returns the acute one as primary and attaches an "ambiguous_note" result showing the obtuse alternative.

Common mistakes

• Using Law of Sines when Law of Cosines is needed. Law of Sines requires a known side-angle pair. For SSS or SAS you must start with Law of Cosines.
• Forgetting the SSA second solution. Real-world problems with measured angles can land in the ambiguous zone; always check whether B₂ = 180° − B₁ also satisfies A + B₂ < 180°.
• Radians vs degrees. All examples assume degree mode. If your manual answer is off by a factor of ~60, you forgot to convert.
• Mixing side-angle labels. Side a is opposite angle A, side b opposite B, side c opposite C. Hand-drawn diagrams sometimes use the wrong pairing.

When to use a different calculator

• For right triangles only, the Right Isosceles Calculator or Special Right Triangles tool is faster.
• For just the area from three sides, Heron's Formula Calculator skips the angle-finding step.
• For coordinate-defined triangles (vertices at (x,y) points), use the Triangle in Coordinate Geometry page.
• For congruence proofs (verifying two triangles match via SSS/SAS/ASA/AAS/HL), see the Congruent Triangle Calculator.

Related concepts

The solver also returns circumradius R = abc / (4·area) — the radius of the circle passing through all three vertices — and inradius r = area / s where s = semi-perimeter. The three altitudes h_a = 2·area / a (similar for h_b, h_c) are also computed. These extras let you verify the triangle quickly: the formula R = abc / (4·area) is independent of the solving method, so a self-consistency check is "did I get the same R both ways?".