Law of Cosines Calculator

The Law of Cosines states that for any triangle with sides a, b, c and opposite angles A, B, C: c² = a² + b² − 2ab cos(C). You can rearrange it for any side: a² = b² + c² − 2bc cos(A) and b² = a² + c² − 2ac cos(B). It generalizes the Pythagorean theorem to all triangles - when C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b².

Use the Law of Cosines for SSS (all three sides known - find all angles) and SAS (two sides and the included angle - find the third side). Use the Law of Sines for SAA/AAS (one side and two angles) and SSA (two sides and a non-included angle, which is the ambiguous case). The mnemonic: if you have a complete side-angle pair plus one more value, the Law of Sines applies; otherwise, use the Law of Cosines.

Rearrange the formula to solve for the cosine of the angle: cos(A) = (b² + c² − a²) / (2bc). Then take the inverse cosine: A = arccos((b² + c² − a²) / (2bc)). This always returns a unique angle between 0° and 180°, so there is no ambiguous case with SSS - each set of valid sides gives exactly one triangle shape.

The triangle inequality states that the sum of any two sides must be greater than the third: a + b > c, a + c > b, and b + c > a. If this is violated, no triangle exists. In the Law of Cosines formula, violating the triangle inequality causes the cosine to exceed 1 in absolute value, making the arccos undefined. This calculator checks for this and reports an error.

The Pythagorean theorem (a² + b² = c²) is a special case of the Law of Cosines when the angle C = 90°. Since cos(90°) = 0, the term 2ab cos(C) vanishes, leaving c² = a² + b². For obtuse angles (C > 90°), cos(C) < 0, so c² > a² + b² - the side opposite the obtuse angle is longer than a Pythagorean hypotenuse would be. For acute triangles, c² < a² + b².

SAS means you know two sides and the angle between them (the included angle). For example: a = 5, b = 8, C = 60° (where C is the angle between sides a and b). The Law of Cosines finds the third side directly: c² = 5² + 8² − 2(5)(8) cos(60°) = 25 + 64 − 40 = 49, so c = 7. Once c is known, the remaining angles follow from further applications of the cosine rule.

Yes - the Law of Cosines works for all triangles: acute, right, and obtuse. For an obtuse angle C (90° < C < 180°), cos(C) is negative, which makes the term −2ab cos(C) positive, correctly producing a larger third side. The formula gives real, valid results for any triangle satisfying the triangle inequality, regardless of whether angles are acute or obtuse.

SSS mode (three sides) uses cos(A) = (b² + c² − a²) / (2bc) to find each angle from the known sides - no ambiguity. SAS mode (two sides and included angle) uses c² = a² + b² − 2ab cos(C) to find the missing side, then finds the remaining angles. SSS solves for angles; SAS solves for the unknown side first, then angles.

The calculator uses the formula Area = ½ × a × b × sin(C), where a and b are two sides and C is the included angle. For SSS mode, C is derived first from the cosine rule, then used in the area formula. This gives the exact area from any complete triangle solution. The formula works because ½ × base × height = ½ × a × (b sin C) when C is the angle at the vertex between sides a and b.

Check three things: (1) all three angles sum to 180°, (2) all sides and angles are positive, (3) the Pythagorean-like check c² ≈ a² + b² − 2ab cos(C) holds for the computed values. You can also verify with the Law of Sines: a/sin(A) should equal b/sin(B) and c/sin(C). A consistent sine ratio confirms the solution.

The Law of Cosines is used extensively in navigation (finding the distance between two GPS points given bearings), surveying (triangulating an inaccessible point), structural engineering (analyzing force triangles), and computer graphics (computing angles in 3D meshes). Any scenario involving three known measurements of a triangle - sides or angles - can be solved with it.