What is the law of cosines?

It generalises the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). It lets you find a side from two sides and the included angle, or an angle from three known sides.

When should I use SSS versus SAS?

Use SSS (three sides) when you know all three side lengths and need the angles. Use SAS (side-angle-side) when you know two sides and the angle between them and need the third side.

Does it work for right and obtuse triangles?

Yes. The law of cosines works for any triangle. For a right angle the cosine term vanishes and it reduces to the Pythagorean theorem.

How do I find an angle when I know all three sides?

Rearrange the rule to cos(A) = (b² + c² − a²) / (2bc), then take the inverse cosine. The calculator does this for each angle automatically in SSS mode.

How is the area of the triangle found?

From two sides and the included angle using Area = ½·b·c·sin(A). This works in both SSS and SAS mode because all three angles are known once the triangle is solved.

What if the three sides cannot form a triangle?

If one side is longer than the sum of the other two, no valid triangle exists and the angle calculation has no real solution. Check that each side is shorter than the sum of the other two (the triangle inequality).

Are angles in degrees or radians?

Inputs and outputs are in degrees. Internally the tool converts to radians for the trigonometric functions, then converts the results back to degrees for display.

Law of Cosines Calculator

The law of cosines is the rule that lets you solve oblique (non-right) triangles where the simple Pythagorean theorem and SOH-CAH-TOA do not apply. Students, engineers, surveyors and anyone working with triangles can use it to find a missing side from two sides and the angle between them, or to recover all three angles from three known sides.

How it works

The core identity is a² = b² + c² − 2bc·cos(A), where each lowercase side sits opposite the matching uppercase angle.

SSS mode (three sides known): the tool rearranges the rule to cos(A) = (b² + c² − a²) / (2bc) and takes the inverse cosine for angles A and B, then finds C as 180° − A − B.
SAS mode (two sides and the included angle): it first computes the missing side a = √(b² + c² − 2bc·cos(A)), then solves for the remaining angles the same way.

In both modes it also returns the area via ½·b·c·sin(A) and the perimeter as a + b + c. The cosine inputs are clamped to the range −1 to 1 to avoid rounding errors at the extremes.

Example

Take a triangle with sides a = 7, b = 8, c = 9 (SSS mode):

Result	Value
Angle A	48.19°
Angle B	58.41°
Angle C	73.40°
Area	26.83
Perimeter	24

Angle A comes from cos(A) = (8² + 9² − 7²) / (2·8·9) = 96/144 = 0.6667, giving A ≈ 48.19°.

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