Triangle Calculator

Mastering Triangle Solutions: Complete Guide

A triangle is defined by three sides and three angles. Given at least three independent pieces of information (with at least one side), the triangle can be solved uniquely (or up to ambiguity in the SSA case). This Professional Triangle Calculator implements all standard solving methods—SSS, SAS, ASA, AAS, and SSA—and provides the complete set of triangle properties including area, perimeter, heights, inradius, and circumradius, accompanied by a scale drawing.

Solving Methods

• SSS (Side‑Side‑Side): All three sides known. Use the Law of Cosines to find each angle.
• SAS (Side‑Angle‑Side): Two sides and the included angle. Law of Cosines gives the third side, then Law of Sines gives the remaining angles.
• ASA (Angle‑Side‑Angle): Two angles and the included side. The third angle is found by the angle sum (180°), then Law of Sines yields the other sides.
• AAS (Angle‑Angle‑Side): Two angles and a non‑included side. Similar to ASA, the third angle is computed first.
• SSA (Side‑Side‑Angle): Two sides and a non‑included angle. This is the ambiguous case; the calculator determines the valid solution(s) and alerts if none exist.

Key Formulas

Law of Cosines: c² = a² + b² − 2ab·cos(C). Used for SSS and SAS.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Used after at least one angle and its opposite side are known.
Heron's Formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.
Heights: hₐ = 2·Area / a, etc.
Circumradius: R = abc / (4·Area).
Inradius: r = Area / s.