Enter any 3 known values (at least one side) and instantly solve the whole triangle — every side, every angle, area, perimeter, heights and radii — with a to-scale diagram. 📐
SSS • SAS • ASA • AAS • SSA
Fill in exactly 3 values (at least one side). Side a is opposite angle A, b opposite B, c opposite C.
Tip: two sides + the angle opposite one of them (SSA) can have two valid triangles — both are shown.
SSS, SAS, ASA, AAS and the tricky ambiguous SSA case — including both solutions when two triangles exist.
A live diagram of your solved triangle with labeled vertices and sides, so you can sanity-check the shape instantly.
Beyond sides and angles: area by Heron’s formula, perimeter, all three heights, inradius and circumradius.
Every triangle is fully determined by three independent measurements, as long as at least one of them is a side. Three angles alone fix only the shape, not the size — there are infinitely many similar triangles with angles 30°-60°-90°. This calculator takes whichever 3 values you know, classifies the case, and solves the rest using two classical tools taught in every US geometry and trigonometry course: the law of sines and the law of cosines.
The law of sines says the ratio of each side to the sine of its opposite angle is constant: a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. It is the natural tool when you know an angle and its opposite side (ASA, AAS, SSA). The law of cosines, c² = a² + b² − 2ab·cos C, generalizes the Pythagorean theorem and handles the cases the law of sines cannot start: three sides (SSS) or two sides with the included angle (SAS). When C = 90°, cos C = 0 and the formula collapses to the familiar c² = a² + b².
Once all three sides are known, the area comes from Heron’s formula: with semi-perimeter s = (a + b + c) / 2, Area = √(s(s−a)(s−b)(s−c)). For a 3-4-5 right triangle, s = 6 and Area = √(6·3·2·1) = 6. From the area, everything else follows: each height is h = 2·Area / base, the inradius is r = Area / s, and the circumradius is R = abc / (4·Area). The calculator also classifies the triangle — equilateral, isosceles or scalene by sides, and acute, right or obtuse by its largest angle.
| Case | You know | Solved with | Solutions |
|---|---|---|---|
| SSS | 3 sides | Law of cosines | 1 (if triangle inequality holds) |
| SAS | 2 sides + included angle | Law of cosines | 1 |
| ASA / AAS | 2 angles + 1 side | Angle sum + law of sines | 1 |
| SSA | 2 sides + non-included angle | Law of sines (ambiguous case) | 0, 1 or 2 |
The SSA “ambiguous case” deserves its reputation. Knowing two sides and an angle opposite one of them is like swinging a door of fixed length toward a wall: it can miss the wall entirely (no triangle), just touch it (one right triangle), or cross it in two places (two distinct triangles). Algebraically, sin B = b·sin A / a may exceed 1 (no solution), equal 1 (one solution) or be less than 1 — giving both an acute B and its supplement 180° − B as candidates. This calculator checks both and displays every valid triangle, which is exactly what trigonometry teachers expect on homework in 2026 — and what surveyors, carpenters and navigators rely on in the field.