Triangle Solver Calculator
Solve any triangle by entering two sides and the included angle. Uses the Law of Cosines to find the missing side — perfect for geometry homework, surveying, and engineering problems.
About this calculator
The Triangle Solver uses the Law of Cosines to find an unknown side when two sides and the included angle are known. The formula is: c = √(a² + b² − 2ab·cos(C)), where a and b are the known sides and C is the angle between them. This generalizes the Pythagorean theorem — when C = 90°, the cosine term vanishes and you recover c = √(a² + b²). Once all three sides are known, remaining angles can be found using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines works for any triangle (acute, obtuse, or right) and is essential in navigation, structural engineering, and physics vector problems.
How to use
Suppose Side A = 7, Side B = 10, and Angle C = 60°. Plug into the formula: c = √(7² + 10² − 2 × 7 × 10 × cos(60°)) = √(49 + 100 − 140 × 0.5) = √(149 − 70) = √79 ≈ 8.888 units. Enter 7 in Side A, 10 in Side B, and 60 in Angle C. The calculator returns Side C ≈ 8.89 units. You can then use the Law of Sines to find the remaining angles A and B if needed.
Frequently asked questions
When should I use the Law of Cosines instead of the Law of Sines?
Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS). The Law of Sines is more convenient when you know two angles and one side (AAS or ASA). The Law of Cosines is also preferred when the Law of Sines could produce an ambiguous case, such as with SSA triangles where two different triangles might satisfy the given conditions.
What is the ambiguous case in triangle solving and how does it affect results?
The ambiguous case (SSA) occurs when you know two sides and an angle that is not between them. Depending on the values, there may be zero, one, or two valid triangles. For example, if the side opposite the given angle is shorter than the other known side, two different triangles can be constructed. Triangle-solving calculators typically flag this situation and return both possible solutions so you can choose the geometrically meaningful one.
How do I find all angles of a triangle once I know all three sides?
With all three sides known (SSS), rearrange the Law of Cosines to solve for each angle: cos(A) = (b² + c² − a²) / (2bc). Apply this formula three times, cycling through the sides, to find all three angles. The results should sum to exactly 180°, which serves as a useful check. This approach is used in GPS triangulation, land surveying, and robotics to determine orientation from measured distances.