Law of Sines and Cosines Calculator (Free Step-by-Step Triangle Solver)

Use this free law of sines and law of cosines calculator to solve any triangle. Choose what you already know — three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), or two sides and a non-included angle (the ambiguous SSA case) — and the tool finds every remaining side and angle, the area, and the perimeter, with a scaled diagram and a clear step-by-step solution.

When do you use the Law of Sines vs. the Law of Cosines?

Use the Law of Cosines when you know all three sides (SSS) or two sides and the angle between them (SAS) — these cases don’t give you a complete side-angle pair to start from. Use the Law of Sines when you already have a matching side and angle pair, which happens with two angles and a side (ASA or AAS) or with two sides and a non-included angle (SSA).

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Law of Sines

In any triangle with sides a, b, c opposite angles A, B, C:  a / sin A = b / sin B = c / sin C.

Law of Cosines

c² = a² + b² − 2ab·cos C  (and the matching forms for a² and b²).

How to use the calculator

1. Pick the case that matches what you know from the “I know” menu (SSS, SAS, ASA, AAS, or SSA).
2. Enter the three known values. Sides are lengths; angles are in degrees.
3. Press Solve to see all sides and angles, the area, the perimeter, a scaled triangle, and the steps.

The ambiguous case (SSA)

When you know two sides and an angle that is not between them, there can be zero, one, or two triangles that fit. The calculator detects this automatically and shows both possible triangles when they exist.

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Looking for more tools? Browse all of our free math calculators and solvers, including the Triangle Calculator and the Trigonometry Calculator.

Frequently asked questions

Does this calculator use the Law of Sines or the Law of Cosines?

Both. It automatically chooses the right one for your case: the Law of Cosines for SSS and SAS, and the Law of Sines for ASA, AAS, and SSA. The steps show exactly which law was applied.

What is the ambiguous case (SSA)?

When you know two sides and a non-included angle, the data can fit zero, one, or two different triangles. The calculator checks for this and displays both valid triangles whenever two solutions exist.

Are the angles in degrees or radians?

All angles are entered and shown in degrees.

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Does it show the area and perimeter?

Yes. Along with every side and angle, it reports the triangle’s area (using one-half the product of two sides and the sine of the included angle) and its perimeter.

Is the Law of Sines and Cosines calculator free?

Yes — it runs in your browser with no sign-up required.

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