Triangle Calculator: Fast and Friendly Tool

This calculator solves a triangle two different ways depending on what you already know: three side lengths (SSS), or two angles plus one side (AAS).

Method 1: three sides (Heron's formula)

Given three side lengths, the calculator computes the semi-perimeter s = (a + b + c) / 2, then the area as √(s(s−a)(s−b)(s−c)). Each angle is found separately with the law of cosines: for example, the angle opposite side a is arccos((b² + c² − a²) / (2bc)).

Worked example: a 3-4-5 triangle

Enter sides 3, 4, 5. Perimeter: 3 + 4 + 5 = 12. Semi-perimeter s = 6. Area = √(6 × 3 × 2 × 1) = √36 = 6.00. Applying the law of cosines to each side gives angles of 36.87°, 53.13°, and 90.00° — a right triangle, since 3-4-5 is the classic Pythagorean triple.

Method 2: two angles and a side (law of sines)

The third angle is found by subtracting the other two from 180° (every triangle's angles sum to 180°). The remaining two sides are then computed from the known side using the law of sines: side = knownSide × sin(angle) / sin(oppositeAngleOfKnownSide).

Validity checks

Before calculating from three sides, the tool checks the triangle inequality — the sum of any two sides must exceed the third (a + b > c, and so on). If it doesn't, no triangle exists with those measurements and the calculator returns "These sides cannot form a triangle" instead of a false result. Similarly, if two entered angles sum to 180° or more, there's no room left for a third angle, and the tool flags the error rather than returning a negative angle.