Triangle Side & Angle Solver

Enter known sides and angles to solve the complete triangle. Uses the Law of Sines and Law of Cosines automatically.

Select Known Values

Enter all 3 sides to find all 3 angles.

Complete Triangle Solution

Triangle
Solved
SSS case

Step-by-Step Solution

How to Solve a Triangle

Solving a triangle means finding all unknown sides and angles. A triangle has six elements: three sides (a, b, c) and three angles (A, B, C). You need at least three elements (with at least one side) to solve a triangle uniquely.

Triangle Solving Cases

SSS (Side-Side-Side)

All three sides are known. Use the Law of Cosines to find each angle.

cos(A) = (b²+c²-a²) / 2bc

SAS (Side-Angle-Side)

Two sides and the included angle are known. Use Law of Cosines for the third side, then Law of Sines or Cosines for the remaining angles.

c² = a²+b²-2ab cos(C)

ASA (Angle-Side-Angle)

Two angles and the included side are known. Find the third angle, then use Law of Sines.

C = 180 - A - B

AAS (Angle-Angle-Side)

Two angles and a non-included side. Find the third angle, then use Law of Sines.

a/sin(A) = b/sin(B)

SSA (Side-Side-Angle)

Two sides and a non-included angle. This is the ambiguous case and may have 0, 1, or 2 solutions.

sin(B) = b sin(A) / a

The Ambiguous Case (SSA)

When two sides and a non-included angle are given, the triangle may not be unique. This is known as the ambiguous case. Depending on the values, there may be no valid triangle, exactly one triangle, or two possible triangles. This calculator detects all cases and shows all valid solutions.

Tips for Solving Triangles

  • Always verify that the angle sum equals 180 degrees.
  • For SSS, check the triangle inequality (sum of any two sides must exceed the third).
  • For SSA, watch out for the ambiguous case with two possible solutions.
  • The largest angle is always opposite the longest side.
  • Use the Law of Cosines when you have SAS or SSS; use the Law of Sines when you have angles.