Solve any triangle using the Law of Cosines. Enter two sides and the included angle (SAS) or three sides (SSS) to find all angles, area, perimeter, and more.
The Law of Cosines is one of the two fundamental tools for solving oblique triangles — triangles that do not have a right angle. It states that for any triangle with sides a, b, c opposite to angles A, B, C respectively: c² = a² + b² − 2ab cos C. This generalises the Pythagorean theorem: when angle C is 90°, cos 90° = 0, and the formula reduces to c² = a² + b².
The Law of Cosines is used in two main configurations. In the SAS (side-angle-side) case, you know two sides and their included angle and need the third side and remaining angles. In the SSS (side-side-side) case, you know all three sides and need all three angles. Once you have the sides and angles, you can compute the area (using ½ab sin C or Heron's formula), the perimeter, the inradius and circumradius, and the three altitudes.
Applications are everywhere — from surveying and navigation (computing distances to unreachable points) to structural engineering (truss analysis) and computer graphics (mesh calculations). This calculator handles both SAS and SSS modes, displays step-by-step solutions, and provides a comprehensive set of derived properties.
This law of cosines triangle solver — sas & sss calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Mode, Angle Unit, Length Unit, Side a and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Law of Cosines: c² = a² + b² − 2ab cos C Angle from sides: C = arccos((a² + b² − c²) / (2ab)) Area (SAS): K = ½ab sin C Area (Heron): K = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2 Inradius: r = K / s Circumradius: R = abc / (4K)
Result: For mode=sss, a=3, b=4, the tool returns the solved law of cosines triangle solver — sas & sss outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in law of cosines triangle solver — sas & sss formulas and reports derived values, checks, and classifications automatically.
This page is tailored to law of cosines triangle solver — sas & sss, with outputs tied directly to the form fields (Mode, Angle Unit, Length Unit, Side a). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
It relates the three sides of a triangle to one of its angles: c² = a² + b² − 2ab cos C. It is a generalisation of the Pythagorean theorem.
Use the Law of Cosines for SAS (two sides + included angle) or SSS (three sides). Use the Law of Sines for AAS or ASA (two angles + any side).
Yes. If cos C is negative, then C > 90°. This happens when c² > a² + b².
When angle C = 90°, cos 90° = 0, so c² = a² + b² − 0 = a² + b². Use this as a practical reminder before finalizing the result.
If any side is greater than or equal to the sum of the other two, the triangle inequality is violated and no triangle exists. Keep this note short and outcome-focused for reuse.
Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. It computes area from three sides alone.