How to Solve a Triangle (Law of Sines & Law of Cosines)

You know two sides and an angle, or two angles and a side, and you need the rest of the triangle. The whole job comes down to one decision: which rule fits what you already know. Get that right and the missing sides and angles fall out in a line or two of algebra. Here’s the decision, case by case.

The setup: sides, angles and how they pair

Label the sides a, b, c and the angle opposite each with the matching capital: A is opposite a, and so on. To pin down a triangle you need three pieces of information, at least one of them a side — because three angles alone fix the shape but leave the size free (all equilateral triangles have three 60° angles, whatever their size).

Law of cosines: SSS and SAS

The law of cosines generalises Pythagoras to any triangle:

c² = a² + b² − 2ab·cos C

• SSS (three sides): rearrange it to find each angle, e.g. cos C = (a² + b² − c²) / (2ab). Do it twice and the third angle is 180° minus the other two.
• SAS (two sides + the included angle): the angle sits between the two known sides, so plug straight in to get the opposite (third) side, then switch to finding the remaining angles.

Use the law of cosines whenever the known angle is between your two known sides, or when you know all three sides.

Law of sines: ASA and AAS

The law of sines says each side is proportional to the sine of its opposite angle:

a / sin A = b / sin B = c / sin C

If you know two angles and any side (ASA = side between the angles, AAS = side outside them), first get the third angle from 180° − (A + B). Now you know all three angles and one side, so the shared ratio gives the other two sides directly.

The ambiguous case: SSA

SSA — two sides and an angle not between them — is the one that trips people up. You’d reach for the law of sines, but there’s a catch: the known side opposite the known angle might be long enough to close the triangle in two different ways, in one way, or not at all.

• Compute sin of the unknown angle from the law of sines.
• If that value is greater than 1, no triangle exists (the side is too short to reach).
• If it’s less than 1, there are usually two candidate angles (an acute one and its obtuse supplement) — check each against the 180° rule; sometimes only one survives.

That’s why a careful solver reports every triangle that fits, rather than silently picking one. The triangle solver does exactly this: enter any three values and it detects the case, applies the right law, and shows both answers when SSA is ambiguous — along with the area and a scaled diagram.

Getting the area too

Once you have two sides and the angle between them, the area is ½·a·b·sin C. From three sides, use Heron’s formula instead. The triangle solver reports the area whichever way you entered the triangle.

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The relationships here — the laws of sines and cosines and the 180° angle sum — are exact identities of Euclidean (flat-plane) geometry. The triangle solver computes them in your browser with standard floating-point trigonometry; nothing is uploaded.