Triangle Calculator

To solve a triangle, you need exactly three known measurements, and at least one of them must be a side length (three angles alone describe infinite similar triangles, not one unique triangle). Depending on which three values you know — three sides (SSS), two sides and the included angle (SAS), or two angles and a side (ASA/AAS) — you apply either the Law of Cosines or the Law of Sines to find the remaining three unknowns. This calculator detects which case applies automatically and solves for all sides, angles, area, and perimeter.

Use the Law of Cosines when you know all three sides (SSS) or two sides and the angle between them (SAS), since it directly relates a side length to the opposite angle: c² = a² + b² − 2ab·cos(C). Use the Law of Sines when you know two angles and any side (ASA or AAS), or after finding one angle from the Law of Cosines, since it states a/sin(A) = b/sin(B) = c/sin(C). In practice, the Law of Cosines solves for missing sides or the first angle, and the Law of Sines is often faster for the remaining angles once one angle is known.

If you know the base and height, Area = (1/2) × base × height. If you know all three sides but not the height, use Heron's Formula: calculate the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). If you know two sides and the included angle, Area = (1/2)ab·sin(C), which is often the fastest method once the triangle has been solved. This calculator applies whichever formula fits the values you entered.

These abbreviations describe which three parts of a triangle you know. SSS means all three sides are known. SAS means two sides and the angle between them (the included angle) are known. ASA means two angles and the side between them are known, while AAS means two angles and a non-included side are known. Each case determines whether you start with the Law of Cosines or the Law of Sines to solve the rest of the triangle.

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be strictly greater than the third side (a+b > c, a+c > b, and b+c > a). If this condition fails — for example sides 2, 3, and 10 — the three lengths simply cannot close into a triangle, since the two shorter sides can't reach far enough to meet. This calculator checks the inequality automatically and flags an error if the sides you entered are impossible.

Yes. Enter any three known values across the six fields (three sides and three angles), including at least one side, and the calculator finds the missing sides and angles automatically. For example, entering two sides and the included angle (SAS) lets the calculator compute the third side using the Law of Cosines, or entering two angles and one side (ASA/AAS) lets it compute the other two sides using the Law of Sines.