Law of Cosines Calculator — Side & Angle Solver

The Formula

The law of cosines generalizes the Pythagorean theorem to any triangle. It relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, c and angle C opposite side c, the formula states that c² equals the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. When C = 90°, cos(90°) = 0 and the formula reduces to c² = a² + b², which is the familiar Pythagorean theorem for right triangles.

Variable Definitions

How to Use This Calculator

1. 1

   Select the mode: "Side (SSS)" when you know all three side lengths, or "Angle (SAS)" when you know two sides and the included angle.

2. 2

   In SSS mode, enter the lengths of all three sides (a, b, c). The calculator finds each angle using the law of cosines step by step.

3. 3

   In SAS mode, enter two side lengths (a, b) and the included angle (C) between them. The calculator finds the third side and the remaining angles.

4. 4

   Review the work steps section to see each calculation step, including cosine evaluations and inverse cosine operations.

5. 5

   Ensure your inputs satisfy the triangle inequality (SSS) or that the given angle is less than 180 degrees (SAS) for a valid triangle.

Quick Reference

Common Applications

• Solving SSS triangles: finding all angles when only side lengths are known, which the law of sines alone cannot do
• Solving SAS triangles: finding the third side and remaining angles when two sides and the included angle are known
• Navigation and GPS: computing distances between points when direct measurement is impossible but distances from two reference points and the included angle are known
• Physics and engineering: calculating resultant vectors using the parallelogram law, where the magnitude of the resultant depends on the angle between two vectors
• Computer graphics: determining polygon geometry and performing collision detection in 3D space where triangle meshes are the fundamental building block

The law of cosines extends the Pythagorean theorem to any triangle. When C = 90 degrees, the formula reduces to the familiar a² + b² = c².

Understanding the Concept

The law of cosines is one of the two fundamental tools (alongside the law of sines) for solving arbitrary triangles. It can be viewed as a generalization of the Pythagorean theorem that includes a correction term for non-right triangles. The formula c² = a² + b² - 2ab·cos(C) expresses the length of side c in terms of sides a, b and the included angle C between them. When angle C is 90 degrees, cos(90°) = 0 and the formula simplifies to c² = a² + b², which is the Pythagorean theorem. When angle C is acute (less than 90 degrees), cos(C) > 0 and the correction term is positive, making c² less than a² + b². When angle C is obtuse (greater than 90 degrees), cos(C) < 0 and the correction term becomes negative, making c² greater than a² + b². This behavior matches the geometric intuition: an obtuse angle "stretches" the opposite side. The law of cosines can be rearranged to solve for angles when all three sides are known: cos(C) = (a² + b² - c²)/(2ab). This rearrangement is the key to solving SSS triangles. The formula can be cyclically permuted for any vertex: a² = b² + c² - 2bc·cos(A) and b² = a² + c² - 2ac·cos(B). The law of cosines has a rich geometric interpretation. It can be derived by dropping an altitude from vertex A to side a, creating two right triangles, and applying the Pythagorean theorem to each. The distance from the foot of the altitude to vertex B is c·cos(B), and to vertex C is b·cos(C). These projections sum to the full side length a, giving the relationship a = b·cos(C) + c·cos(B), which is known as the projection formula and is equivalent to the law of cosines. Numerically, the law of cosines is well-behaved for most triangles but can be less accurate for very flat triangles (where angles are near 0 or 180 degrees) due to floating-point precision in the arccosine function. In such cases, the law of sines may give more accurate angle results after the first angle is determined.

Frequently Asked Questions

Sources & References