Law of Cosines Calculator – Triangle Solver
Solve any triangle using the Law of Cosines. Find missing sides and angles with precision.
Included Angle (C)
Calculation Results
What is the Law of Cosines?
The Law of Cosines is a powerful trigonometric theorem used to calculate the missing sides or angles of any triangle, not just right-angled ones.
The Master Formula
The primary equation is c² = a² + b² - 2ab × cos(C). It relates the lengths of the sides of a triangle to the cosine of one of its angles.
Pythagorean Connection
If angle C is exactly 90 degrees, cos(90°) is 0. The formula simplifies to c² = a² + b², making it a generalized version of the Pythagorean theorem.
SAS Condition
Use this law when you know two sides and the included angle (Side-Angle-Side) and you need to calculate the length of the unknown third side.
SSS Condition
You can also use it when you know all three side lengths (Side-Side-Side) and you need to find the measure of any of the triangle's internal angles.
Inverse Cosine
When finding an angle, the formula is rearranged to cos(C) = (a² + b² - c²) / 2ab. You then use the arccos function to find the angle.
No Ambiguous Cases
Unlike the Law of Sines, the Law of Cosines does not suffer from the "ambiguous case" where an angle could be acute or obtuse. The cosine value dictates it perfectly.
Where is the Law of Cosines Used?
Trigonometry extends far beyond the classroom. These geometric principles are applied daily in various professional and scientific fields.
Surveying & Land Mapping
Surveyors use the theorem to find the distance across impassable terrain, like a lake or canyon, by measuring two accessible distances and the angle between them.
Navigation & Sailing
If a ship changes course by a specific angle after traveling a certain distance, navigators calculate how far they are from the starting point using this law.
Architecture & Engineering
Engineers use non-right triangle calculations to determine the required lengths of beams in roof trusses, suspension bridges, and structural supports.
Robotics & Kinematics
In robotic arm design (inverse kinematics), engineers calculate the exact joint angles required to position a robotic hand at a specific 3D coordinate.
Aviation & Flight Paths
Pilots use vectors and triangle solving to calculate crosswind corrections, determining their true heading and ground speed during a flight.
Astronomy
Astronomers apply spherical variations of the Law of Cosines to calculate the apparent distances between stars and planets as seen from Earth.
How to Avoid Calculation Mistakes?
Ensure perfect accuracy in your geometry homework and real-world projects by keeping these crucial tips in mind.
Order of Operations
A common calculator error is subtracting 2ab from a²+b² before multiplying by cos(C). Remember PEMDAS: multiplication happens before subtraction!
Check Degree Mode
If your angle is in degrees, ensure your calculator is set to 'Degree' mode, not 'Radian'. Using the wrong mode will result in wildly incorrect answers.
Triangle Inequality
When entering three sides (SSS), ensure the sum of any two sides is greater than the third side. Otherwise, a triangle cannot physically exist.
Negative Cosines
Don't panic if cos(C) is negative! This simply means the angle C is obtuse (greater than 90°). The formula handles this naturally.
Don't Forget to Square Root
The formula calculates c², not c. After doing all the math on the right side of the equation, you must take the square root to find the actual side length.
Switching to Law of Sines
Once you use the Law of Cosines to find one missing piece, it's often faster to switch to the Law of Sines (a/sinA = b/sinB) to find the remaining angles.
