Triangle Solver

Three pieces of information are enough because a triangle has three degrees of freedom: the three side lengths, or any combination that adds up to the same amount of independent data. The three standard combinations have short names used in every geometry class. SSS means you measured all three sides - common when the triangle is an object you can walk around with a tape measure. SAS means two sides and the angle where they meet, which is what you get from a bevel gauge or a theodolite reading between two visible corners. ASA means two angles and one connecting side, the typical result of measuring a baseline and two bearings from its ends.

Once you have three valid values, the rest follows with two rules. The law of cosines turns three sides into an angle: a² = b² + c² − 2bc·cos A, rearranged as cos A = (b² + c² − a²) / (2bc) when the angle is what you need. The law of sines then passes the problem around the triangle: a / sin A = b / sin B = c / sin C, so one known angle-side pair unlocks the other two. The third angle costs nothing at all - subtract the two you know from 180°.

Area has two equally good formulas, and the solver picks the one that matches your inputs. With two sides and the included angle it is area = ½·a·b·sin C. With three sides it is Heron's formula: halve the perimeter to get s, then take √(s(s−a)(s−b)(s−c)). Both give the answer in square units of whatever unit you typed for the sides, so centimetres become square centimetres without any conversion step.

Watch for two failure modes. Three lengths that violate the triangle inequality - any side equal to or longer than the other two added together - collapse into a flat line or nothing at all, and the solver says so instead of returning nonsense. In the two-angle case, the two known angles must add up to less than 180°; if they reach or pass it, there is no room left for the third angle. Small rounding errors in field measurements usually show up as angles that miss 180° by a fraction of a degree, which is a sign to re-measure rather than trust the numbers.

Where this comes in handy: checking a rafter or truss cut against the plan angle, confirming that a bracket really is square, converting a bearing into a corner angle for trim work, or solving the triangles hidden inside a ramp, a roof plane or a solar array layout. If one of your values came from the on-screen protractor instead of a tape, type it in as an angle and let the solver work out the geometry around it.