Triangle Calculator

About Triangle Calculations

A triangle is fully determined when three independent measurements are known. The five common cases — SSS, SAS, ASA, AAS, and SSA — each use different laws to solve for the remaining sides and angles. The Law of Cosines (c² = a² + b² − 2ab·cos(C)) handles cases with at least two sides and a contained angle. The Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) applies when angles and their opposite sides are related.

Heron's formula computes the area from all three sides using the semi-perimeter: s = (a+b+c)/2, then A = √(s(s-a)(s-b)(s-c)). This is useful when no height is directly known. Once you have three sides, you can always compute all angles using the Law of Cosines and then verify using Heron's formula.

The SSA case (two sides and a non-included angle) is called the ambiguous case because it can produce zero, one, or two valid triangles. If the given side opposite the angle is shorter than the perpendicular height from the vertex, no triangle exists. If it equals the perpendicular height, exactly one right triangle exists. If it's between the perpendicular height and the other given side, two triangles are possible.