Law of Sines & Cosines Calculator
Solve any triangle from a valid mix of sides and angles. The tool chooses the Law of Sines or the Law of Cosines for you and flags the ambiguous SSA case.
Solved triangle
Two laws, one solver
Right-triangle trig only reaches so far — most triangles have no 90° angle. The Law of Sines relates each side to the sine of its opposite angle, so it solves triangles where you already know a side and its partner angle. The Law of Cosines generalises the Pythagorean theorem, so it handles the cases the Law of Sines can’t: two sides with the angle between them, or all three sides.
Enter what you know using the a/b/c and A/B/C fields, leave the unknowns blank, and the solver reports which method it used along with the full set of sides, angles, and the area.
Frequently Asked Questions
When do I use the Law of Sines versus the Law of Cosines?
Use the Law of Sines when you have a matching side–angle pair — that covers ASA, AAS, and the SSA case. Use the Law of Cosines when you don't: two sides and the included angle (SAS), or all three sides (SSS). This tool detects which situation you've entered and applies the right law for you.
What is the ambiguous (SSA) case?
When you know two sides and an angle opposite one of them, there can be zero, one, or two valid triangles. This is the ambiguous case. The calculator returns the primary, acute-angle solution and warns you when a second triangle with an obtuse angle also fits your numbers, so you can decide which one your problem intends.
Which side goes with which angle?
By convention, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Keeping that pairing straight is the single most important step — the laws relate each side to the angle across from it, so mislabeling them is the most common source of a wrong answer.
How is the area calculated?
Once the triangle is fully solved, the area is found with the formula one-half times two sides times the sine of the angle between them. Because the tool already knows all three sides and angles, it can use any consistent pair, and the result is the same either way.
Educational tool only. Results are trigonometric estimates rounded for display. In the ambiguous SSA case a second valid triangle may exist — check which solution your problem intends.