Triangle Solver
Enter any three measurements and get the other three, the area, a drawing to scale, and the step-by-step work — Pythagorean theorem, Law of Sines, or Law of Cosines, whichever the problem actually needs.
Most triangle problems come down to one question: which rule applies here? Type in what you know and this tool names the case (SSS, SAS, ASA, AAS, or the ambiguous SSA), shows the law it used at each step, and draws the triangle to scale so you can see whether your answer is plausible. When SSA has two valid answers — the case that quietly costs the most points — you get both.
Side a is always opposite angle A.
Try a case:
Which rule do I use? The whole decision tree
- SSS — three sides. Law of Cosines for the first two angles, then subtract from 180°. First check the triangle inequality: the two shorter sides together must be longer than the longest one, or no triangle exists.
- SAS — two sides and the angle between them. Law of Cosines gives the third side, then keep using it for the next angle. Always exactly one answer.
- ASA or AAS — two angles and any side. The third angle is 180° minus the other two, then Law of Sines for the sides. Always exactly one answer.
- SSA — two sides and an angle not between them. This is the ambiguous case. Law of Sines gives sin B, and two different angles share the same sine: one acute and one obtuse. Depending on the numbers you get two triangles, one, or none. This is the case textbooks test and students miss.
- AAA — three angles. Not solvable. Three angles fix the shape but not the size, so infinitely many similar triangles fit. You need at least one side.
- The Pythagorean theorem is the SSS and SAS case with a 90° angle: a² + b² = c² where c is the hypotenuse. Enter two legs and 90° for the angle between them, or enter all three sides to check whether a triangle is right.
- Why the Law of Cosines for angles, not the Law of Sines? The inverse sine always returns an acute angle, so an obtuse angle comes back wrong. The inverse cosine tells acute from obtuse correctly. This tool uses cosines wherever the angle could be obtuse.
- Typing: angles are in degrees; fractions like
3/4and commas in1,250are fine. Units cancel out, so use inches, cm, or anything else — the answer is in the same unit you typed. Area comes out in those units squared.
Everything runs in your browser. What you type is saved on this device only.