Solve a triangle using the law of cosines — find a missing side from two sides and the included angle, or find an angle from three sides.
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How to use
Enter two sides (a and b) and the included angle C to find side c and solve the full triangle.
Also in Triangles
Trigonometry
The law of cosines calculator solves a triangle when you know three sides (SSS) or two sides and the included angle (SAS). It generalises the Pythagorean theorem to non-right triangles and returns all missing sides, angles, and the area.
The law of cosines relates the three sides of any triangle to one of its angles. It is most useful when you have two sides and the angle between them (SAS) or all three sides (SSS). For a right triangle the formula reduces to the Pythagorean theorem because cos(90) is zero.
c^2 = a^2 + b^2 - 2ab * cos(C)
Finds side c from sides a, b and included angle C.
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Finds angle C from all three sides.
Use SAS mode when you know two sides and the angle between them. Use SSS mode when you know all three sides and need the angles. For other configurations such as ASA or AAS, the law of sines is usually more convenient.
When angle C is exactly 90 degrees, cos(C) equals zero, so the formula simplifies to c squared equals a squared plus b squared. The law of cosines is therefore a generalisation that works for all triangles, not just right triangles.
Frequently asked questions
Yes. Rearrange the formula to solve for each angle in turn: cos(A) = (b^2 + c^2 - a^2) / (2bc), and similarly for B and C.
cos(90) = 0, so the cross term vanishes and the formula becomes the Pythagorean theorem.
Yes. "Law of cosines" and "cosine rule" are two names for the same formula.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Solve a triangle using the law of sines — find missing sides or angles from a known side-angle pair and one additional measurement.
Solve for the missing side of a right triangle using the Pythagorean theorem, with perimeter and area included.
Calculate sine, cosine, and tangent ratios for a right triangle from side lengths or find missing sides from an angle.
Calculate triangle area from base and height, three sides, two sides and an included angle, or coordinates.
