Law of Cosines
c² = a² + b² − 2ab cos C generalises Pythagoras.
About this topic
The Law of Cosines, c² = a² + b² − 2ab cos C, generalises the Pythagorean theorem to any triangle; when C = 90° the last term vanishes. Use it when you know two sides and the included angle (SAS) or all three sides (SSS) to find an angle.
Rearranged: cos C = (a² + b² − c²)/(2ab). If the result is positive the angle is acute, if zero it is a right angle, and if negative the angle is obtuse. Unlike the Law of Sines, the cosine formula gives an unambiguous angle between 0° and 180°, so it is the safer choice when an obtuse angle might be involved. When finding all angles, compute the largest one first.
Example: two sides of 7 and 9 meet at 50°. The third side is c² = 49 + 81 − 2(7)(9)cos 50° = 130 − 126(0.643), about 49, so c is about 7.0. Ships leaving a port on different bearings, or a robot arm with two segments, use this to find the distance between endpoints.
Tips: pair the angle with the side opposite it, do the multiplication before the subtraction (the formula is not c² = (a² + b² − 2ab) cos C), and use radians or degrees consistently.
Key ideas
- Use with SAS or SSS
- Reduces to Pythagoras when C = 90°
- Solve for angles with rearrangement
Quick check
Use cases: where Law of Cosines helps
- Homework: review the key idea behind Law of Cosines before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Law of Cosines without notes.
- Visual learning: play with the interactive lab to see how Law of Cosines changes when you adjust its inputs.
- Teaching a friend: explain Law of Cosines in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Law of Cosines?
c² = a² + b² − 2ab cos C generalises Pythagoras.
What will I learn in Law of Cosines?
Use with SAS or SSS. Reduces to Pythagoras when C = 90°. Solve for angles with rearrangement.
Is Law of Cosines part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Law of Cosines lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Law of Cosines?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.