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Triangle Solver — SSS, SAS, ASA, AAS

Solve any triangle using law of cosines and law of sines. Returns all sides, angles, area, and perimeter.

Triangle formulas

Angle sumA + B + C = 180°
Law of sinesasin A = bsin B = csin C
Law of cosinesc2 = a2 + b2 − 2ab cos C
Angle from three sidescos C = a2 + b2 − c22ab
Area (SAS)Area = ½ ab sin C
Heron's formulaArea = √s(s − a)(s − b)(s − c), s = a + b + c2

Where Side a is opposite angle A, b opposite B, c opposite C

Worked examples

  1. SSS: sides 7, 8 and 9. Find the angles and area.

    1. cos C = 49 + 64 − 812 × 7 × 8 = 32112 → C ≈ 73.40°
    2. cos A = 64 + 81 − 492 × 8 × 9 → A ≈ 48.19°
    3. B = 180° − 73.40° − 48.19° = 58.41°
    4. s = 12, Area = √12 × 5 × 4 × 3 = √720

    Answer: A ≈ 48.19°, B ≈ 58.41°, C ≈ 73.40°, Area ≈ 26.83

  2. ASA: A = 40°, B = 60°, side c = 10. Find the other sides.

    1. C = 180° − 40° − 60° = 80°
    2. a = 10 × sin 40°sin 80° ≈ 6.527
    3. b = 10 × sin 60°sin 80° ≈ 8.794

    Answer: C = 80°, a ≈ 6.53, b ≈ 8.79

Tips & common mistakes

  • SSA (two sides and a non-included angle) can have two solutions — the "ambiguous case".
  • Use the law of cosines first for SSS and SAS, then the law of sines for the rest.
  • The longest side is always opposite the largest angle.