Triangle Solver
Solve triangles using angles and sides with the law of sines, law of cosines, and trigonometric functions
Side a
Side b
Side c
Angle A
Angle B
Angle C
Area
Perimeter
Type
Second Solution (Ambiguous Case)
Side a
Side b
Side c
Angle A
Angle B
Angle C
Area
Perimeter
Type
Solution Steps
Triangle Solving Formulas
Law of Cosines
Used to find a side when two sides and the included angle are known (SAS), or to find an angle when all three sides are known (SSS).
a² = b² + c² - 2bc · cos(A)
b² = a² + c² - 2ac · cos(B)
c² = a² + b² - 2ab · cos(C)
Law of Sines
Used to find unknown sides or angles when certain pairs of sides and angles are known (ASA, AAS, SSA).
a / sin(A) = b / sin(B) = c / sin(C)
Area Formulas
Area = ½ · a · b · sin(C)
Area = √[s(s-a)(s-b)(s-c)] (Heron's formula)
where s = (a + b + c) / 2 (semi-perimeter)
Triangle Solving Cases
SSS (Side-Side-Side)
When all three sides are known, the Law of Cosines is used to find each angle. This always produces exactly one unique triangle if the triangle inequality is satisfied.
SAS (Side-Angle-Side)
When two sides and the included angle are known, the Law of Cosines finds the third side, then the Law of Sines finds the remaining angles. Always produces one unique triangle.
ASA (Angle-Side-Angle)
When two angles and the included side are known, the third angle is found by subtraction, then the Law of Sines finds the remaining sides. Always produces one unique triangle.
AAS (Angle-Angle-Side)
When two angles and a non-included side are known, the third angle is found by subtraction, then the Law of Sines finds the remaining sides. Always produces one unique triangle.
SSA (Side-Side-Angle) — Ambiguous Case
When two sides and a non-included angle are known, there may be zero, one, or two valid solutions. The Law of Sines may produce two possible values for the unknown angle (since sin is positive in both the first and second quadrants). This calculator checks for and displays all valid solutions.
Triangle Classification
By Sides
- Equilateral — All three sides are equal (all angles = 60°)
- Isosceles — Exactly two sides are equal
- Scalene — All three sides are different
By Angles
- Acute — All angles are less than 90°
- Right — One angle is exactly 90°
- Obtuse — One angle is greater than 90°
References
The formulas and methods used in this triangle solver are based on established trigonometric principles from authoritative sources:
Related Calculators
This calculator provides mathematically exact results based on the given inputs. Rounding is applied only for display purposes (up to 6 decimal places). For the SSA (ambiguous) case, always verify which solution applies to your specific geometric context.
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