Calculate unknown sides and angles of any triangle using the Law of Cosines. Supports both SAS (Side-Angle-Side) and SSS (Side-Side-Side) calculations with step-by-step solutions.
Geometry often demands precision, especially when solving triangles. The Law of Cosines Calculator provides an efficient way to determine unknown sides or angles, particularly in non-right triangles. Instead of manually applying formulas, this calculator delivers instant, step-by-step solutions. Whether you are a student, engineer, surveyor, or math enthusiast, understanding and applying the cosine rule can make complex problems much easier.
The law of cosines (also called the cosine rule) is a fundamental trigonometric identity used to relate the sides and angles of a triangle. Its general form is:
a² = b² + c² – 2bc cos(A)
Where:
a, b, c = sides of the triangle
A = angle opposite side a
Other versions include:
b² = a² + c² – 2ac cos(B)
c² = a² + b² – 2ab cos(C)
This formula works for all triangles: acute, obtuse, or right-angled. In fact, for a right triangle, the cosine rule simplifies to the Pythagorean Theorem.
If you often deal with vertex-based calculations or coordinate geometry, you may also find the Vertex Calculator highly useful.
A Law of Cosines Calculator uses your inputs—two sides and the included angle (SAS) or three sides (SSS)—to determine unknown values.
Step-by-step process:
Enter known values (sides and/or angle).
The calculator applies the cosine rule formula.
Missing side lengths or angles are computed.
It may also provide the perimeter and area of the triangle.
Benefits:
Saves time compared to manual calculations.
Reduces errors in applying trigonometric formulas.
Ideal for both academic learning and professional use.
If you’re analysing relationships between different geometric properties, try the Critical Points Calculator for a deeper understanding of curve behaviour.
The law of cosines applies when:
You know two sides and the included angle (SAS) → Find the third side.
You know all three sides (SSS) → Find any angle.
You need to check if a triangle is obtuse, right, or acute.
You want to validate triangle properties without ambiguity.
Examples:
SAS: Given sides b = 7, c = 9, and angle A = 60°, find side a.
SSS: Given sides a = 5, b = 6, c = 7, find angle A.
To visualise range-based problems, you might also explore the Interval Notation Calculator, which helps in expressing value intervals precisely.
Beyond classrooms, the cosine rule is widely used in practical fields such as:
Navigation & GPS: Calculating distances between points on Earth using the spherical law of cosines.
Surveying & Mapping: Measuring inaccessible distances across rivers or mountains.
Engineering & Architecture: Structural analysis and designing triangular frameworks.
Astronomy: Determining distances between celestial bodies.
Physics: In vector addition, forces, and projectile analysis.
Computer Graphics & Robotics: Calculating rotations, paths, and transformations.
If you work with measurement precision, the Percent Error Calculator can help evaluate experimental accuracy efficiently.
While the basic law applies to plane triangles, there are important extensions:
Spherical Law of Cosines:
cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)
Used in astronomy, navigation, and geodesy.
Vector Dot Product Connection:
The dot product definition leads directly to the cosine rule, showing its foundation in vector mathematics.
Law of Cosines in 3D Geometry:
Useful in physics and engineering to find angles between 3D vectors.
Relation to Law of Sines:
Complements the cosine rule for solving all types of triangles.
Example 1: Solving a Triangle (SAS Case)
Given sides b = 8 cm, c = 6 cm, and angle A = 45°, find side a.
Solution:
a² = b² + c² – 2bc cos(A)
a² = 8² + 6² – 2(8)(6)cos(45°)
a² = 64 + 36 – 96(0.707)
a² ≈ 32.1
a ≈ 5.67 cm
Example 2: Finding an Angle (SSS Case)
Given a = 10, b = 7, c = 12, find angle A.
Solution:
cos(A) = (b² + c² – a²) / (2bc)
cos(A) = (49 + 144 – 100) / (168)
cos(A) = 93 / 168 ≈ 0.553
A ≈ 56.4°
For interpolation-based estimations or continuous datasets, you can also check the Interpolation Calculator.
Law of Sines: Best when you know an angle and its opposite side (ASA, AAS). The law of cosines is stronger when dealing with SAS or SSS cases.
Pythagorean Theorem: Limited to right triangles. The cosine rule works for all triangles, including obtuse ones.
Triangle Solver Calculators: Many tools combine sine and cosine laws. However, a dedicated law of cosines calculator ensures more accurate solutions for ambiguous triangles.
Confusing angle placement: Always match the side with its opposite angle.
Ignoring units: Ensure consistency (cm, m, km).
Radians vs degrees: Enter the correct mode in calculators.
Rounding errors: Keep enough decimal places until the final step.
Wrong case choice: Don’t apply cosine rule when sine rule is simpler.
Skipping diagram drawing: A sketch often prevents misinterpretation.
Misusing cosine values for obtuse angles: Remember cos(>90°) is negative.
The law of cosines is not modern—it dates back centuries:
Euclid: Early geometric principles laid its foundation.
Al-Kashi (15th century): Provided one of the earliest general formulas for any triangle, earning the law another name—Al-Kashi’s Theorem.
Astronomy & Navigation: Ancient explorers used spherical variations to chart stars and seas.
Today, computer science, physics, and engineering rely heavily on its vector connections. Understanding this history not only deepens appreciation but also connects modern learners to centuries of mathematical discovery.
Critical Points Calculator – Understand turning points in functions.
Vertex Calculator – Find the vertex of parabolas effortlessly.
Interval Notation Calculator – Express intervals in standard notation.
Interpolation Calculator – Estimate values within data points.
Percent Error Calculator – Measure accuracy in experiments.
Distributive Property Calculator – Simplify expressions with step-by-step expansion.
Yes, it works for all types—acute, obtuse, or right-angled.
The law of cosines is ideal for SAS and SSS, while the law of sines is best for ASA and AAS cases.
Absolutely, but be cautious since cosine values turn negative in obtuse cases.
A variation used in navigation and astronomy to calculate distances on a sphere, such as Earth.
Yes, it’s essential in vector problems, particularly when calculating the angle between forces.
The Law of Cosines Calculator transforms complex triangle problems into quick, accurate solutions. From classrooms to construction sites, it’s a versatile tool bridging theoretical maths and real-world applications. Explore our other geometry and trigonometry calculators to continue learning, visualising, and mastering mathematical relationships with confidence.