Triangle Solver Tutorial — SSS, SAS, ASA, AAS, SSA Methods Explained

“Solving a triangle” means: given just three of the six parts of a triangle (three sides + three angles), find the other three. The exact method depends on which three you have. Five named cases cover every solvable combination: SSS, SAS, ASA, AAS, SSA. This guide walks each one with the formulas you need and a worked example, then explains why SSA is “ambiguous” and how to handle it.

The Six Parts of a Triangle

Every triangle has 6 measurable parts: three sides (usually labelled a, b, c) and three angles (A, B, C — each opposite the side of the same letter). You only need 3 of them — provided at least one is a side — to solve the rest. The 5 valid “given” combinations are the methods below.

The Two Master Formulas

All five methods reduce to one of these two relationships:

• Law of Cosines: c² = a² + b² − 2ab × cos(C)
  Solves for a side when you have two sides + 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 + one more angle, or solves for an angle when you have two sides + one opposite angle.

Method 1 — SSS (Three Sides)

When to use: You know all three side lengths a, b, c.
Steps:

1. Use Law of Cosines to find any one angle: cos(C) = (a² + b² − c²) / (2ab) → C = arccos(…)
2. Use Law of Sines to find a second angle.
3. Third angle = 180° − (sum of first two).

Example: a = 5, b = 7, c = 9. Find all three angles.

• cos(C) = (25 + 49 − 81) / (2 × 5 × 7) = −7 / 70 = −0.1 → C ≈ 95.74°
• sin(A) / 5 = sin(95.74°) / 9 → sin(A) ≈ 5 × 0.9950 / 9 ≈ 0.5528 → A ≈ 33.56°
• B = 180° − 95.74° − 33.56° = 50.70°

SSS always gives a unique triangle (provided the triangle inequality a + b > c holds for all three pairs).

Method 2 — SAS (Two Sides + Included Angle)

When to use: You know two sides and the angle between them (e.g. a, b, C).
Steps:

1. Law of Cosines to find the missing side opposite the known angle: c² = a² + b² − 2ab × cos(C)
2. Law of Sines to find another angle.
3. Third angle = 180° − sum of the other two.

Example: a = 8, b = 10, C = 60°. Find c, A, B.

• c² = 64 + 100 − 2(8)(10) cos(60°) = 164 − 160 × 0.5 = 84 → c ≈ 9.17
• sin(A) / 8 = sin(60°) / 9.17 → sin(A) ≈ 8 × 0.8660 / 9.17 ≈ 0.7558 → A ≈ 49.11°
• B = 180° − 60° − 49.11° = 70.89°

SAS always gives a unique triangle.

Method 3 — ASA (Two Angles + Included Side)

When to use: You know two angles and the side between them (e.g. A, B, c).
Steps:

1. Third angle = 180° − A − B.
2. Law of Sines to find each of the other sides.

Example: A = 50°, B = 60°, c = 12. Find C, a, b.

• C = 180° − 50° − 60° = 70°
• a / sin(50°) = 12 / sin(70°) → a = 12 × 0.766 / 0.9397 ≈ 9.78
• b / sin(60°) = 12 / sin(70°) → b = 12 × 0.866 / 0.9397 ≈ 11.06

ASA always gives a unique triangle (any two angles summing to less than 180° + any positive side define one triangle).

Method 4 — AAS (Two Angles + a Non-Included Side)

When to use: You know two angles and a side that is not between them (e.g. A, B, a).
Steps: Same as ASA — compute the third angle, then Law of Sines for the remaining sides. The only difference from ASA is the position of the known side (here it’s opposite one of the known angles).

Example: A = 45°, B = 65°, a = 7. Find C, b, c.

• C = 180° − 45° − 65° = 70°
• b / sin(65°) = 7 / sin(45°) → b = 7 × 0.9063 / 0.7071 ≈ 8.97
• c / sin(70°) = 7 / sin(45°) → c = 7 × 0.9397 / 0.7071 ≈ 9.30

Method 5 — SSA (The Ambiguous Case)

When to use: You know two sides and an angle opposite one of them (not between — e.g. a, b, A).
Why “ambiguous”: SSA can produce zero, one, or two valid triangles depending on the specific values. This is the only case requiring case-checking.

Steps to handle SSA:

1. Law of Sines to find the angle opposite the other known side: sin(B) = b × sin(A) / a
2. If sin(B) > 1 → no triangle exists (the given side is too short to reach).
3. If sin(B) = 1 → exactly one right triangle (B = 90°).
4. If sin(B) < 1 → two candidates: B₁ = arcsin(…), B₂ = 180° − B₁. Both might give valid triangles if A + B₂ < 180°.
5. For each valid B, finish via ASA: C = 180° − A − B, then c via Law of Sines.

Example (two solutions): a = 6, b = 8, A = 35°. Find B, C, c.

• sin(B) = 8 × sin(35°) / 6 = 8 × 0.5736 / 6 ≈ 0.7648
• B₁ ≈ 49.86°, B₂ = 180° − 49.86° = 130.14°
• Check B₂: A + B₂ = 35° + 130.14° = 165.14° < 180° → both valid
• Triangle 1: C = 180° − 35° − 49.86° = 95.14°, c = 6 × sin(95.14°) / sin(35°) ≈ 10.41
• Triangle 2: C = 180° − 35° − 130.14° = 14.86°, c = 6 × sin(14.86°) / sin(35°) ≈ 2.68

This is why textbooks warn about SSA: real-world problems with measured angles can land in the ambiguous zone, and you need geometric context (e.g. “the shortest possible triangle”) to pick the right solution.

Diagnostic Flowchart — Which Method Do I Use?

1. Count what you’re given.
2. If all 3 are sides → SSS
3. If 2 sides + 1 angle:
   • Angle is between the two sides → SAS
   • Angle is opposite one of the sides (not between) → SSA (check for ambiguity)
4. If 2 angles + 1 side:
   • Side is between the two angles → ASA
   • Side is opposite one of the angles → AAS
5. If 3 angles (AAA) → infinite similar triangles, no unique solution. AAA defines shape but not size; you need at least one side.

Common Mistakes

• Using Law of Sines when Law of Cosines is needed — Law of Sines requires a side-angle pair to be opposite each other. For SSS or SAS, you must start with Law of Cosines.
• Forgetting the SSA ambiguous second solution — always check whether B₂ = 180° − B₁ also satisfies A + B₂ < 180°.
• Confusing radians vs degrees on your calculator — every example above assumes degree mode. If your answer is “wildly wrong by a factor of ~60”, you’re in radian mode.
• Mixing up side ↔ opposite angle pairings — side a is opposite angle A, not angle a. A common labelling slip on hand-drawn figures.
• Thinking SSA means “no possible triangle” — SSA doesn’t always fail; it just requires case-checking. SSS, SAS, ASA, AAS are always unambiguous.

FAQ

What’s the difference between ASA and AAS? The position of the known side. In ASA the side is between the two known angles; in AAS it’s opposite one of them. Both always give a unique triangle, but the formula sequence differs slightly (in AAS you still find the third angle first via 180° − sum, then apply Law of Sines).

Why isn’t there an “SSS Law of Sines” method? Law of Sines needs a side-angle pair where the angle is opposite the side. With pure SSS you have no angles to start the chain — so you must use Law of Cosines first to extract one angle, then switch to Law of Sines.

Can the triangle solver handle right triangles? Yes — SOH-CAH-TOA + Pythagoras are special cases of these general formulas. When C = 90°, Law of Cosines reduces to c² = a² + b² (Pythagoras), and Law of Sines reduces to sin(A) = a/c (CAH).

What if I have one side and two angles, but neither angle is opposite the side? That’s still solvable — it’s just AAS with the angles relabelled. Compute the third angle (180° − sum), and now one of the original angles is opposite the known side, letting you proceed.

To skip the manual work, use our Triangle Solver — enter any 3 valid parts and it returns the other 3 instantly, plus the step-by-step working using the methods above. For congruence proofs that use these same postulates, see How to Prove Two Triangles Are Congruent. For solving for x when one of the inputs is an algebraic expression, see How to Find x in Geometry Problems. The full formula reference is on the Triangle Formulas page. For tricky non-standard problems (figure given as a photo, or 3D extensions), upload to the AI Math Solver.