How to solve a triangle
A triangle has six measurements: three sides (a, b, c) and three angles (A, B, C). Side a is opposite angle A, and so on. Any three of them, with at least one side, are usually enough to find the rest. Which rule you use depends on which three you know.
| You know | Name | Start with |
|---|---|---|
| Three sides | SSS | Law of cosines for an angle |
| Two sides and the angle between them | SAS | Law of cosines for the third side |
| Two angles and the side between them | ASA | Angle sum, then law of sines |
| Two angles and another side | AAS | Angle sum, then law of sines |
| Two sides and an angle not between them | SSA | Law of sines (check for two solutions) |
Law of cosines
c² = a² + b² − 2ab·cos C. It generalizes the Pythagorean theorem: when C = 90°, the cosine term vanishes and you get c² = a² + b².
Law of sines
a / sin A = b / sin B = c / sin C. Each side divided by the sine of its opposite angle gives the same number, which equals the diameter of the circumscribed circle.
The ambiguous case (SSA)
If you know two sides and an angle that is not between them, the law of sines gives a value for sin of the second angle. Because sin θ = sin(180° − θ), both an acute and an obtuse angle may fit. The solver checks each one and shows every triangle that works. Try the SSA, two triangles example to see both.
FAQ
Can I solve a right triangle here?
Yes. Enter 90 for the right angle and any two other values. For example, legs a = 3 and b = 4 with C = 90 gives hypotenuse c = 5.
Why does the solver say no triangle exists?
Either the sides break the triangle inequality (the two shorter sides must add up to more than the longest), the angles add up to 180° or more, or in the SSA case the given side is too short to close the triangle.
Can I use the diagram in a worksheet?
Yes. Press Download PNG to save a high-resolution, to-scale image with all labels.