Law of Cosines in Geometry
Generalised Pythagorean theorem.
Key idea
Generalised Pythagorean theorem.
In depth
c² = a² + b² − 2ab cos C generalises Pythagoras by correcting for a non-right angle. It lets you find a side from two sides and the angle between them, or an angle from three sides.
Worked example
a = 5, b = 7, C = 60°: c² = 25 + 49 − 35 = 39, c ≈ 6.24.
Formula
Real-world uses
- Finding the third side of a triangular plot from two sides and the angle between them
Common mistake
Using the angle that is not between the two known sides.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Law of Cosines in Geometry?
Generalised Pythagorean theorem. It is a topic at the advanced level in Geometry.
What is the formula for Law of Cosines in Geometry?
The key formula is c² = a² + b² − 2ab cosC.
Where is Law of Cosines in Geometry used in real life?
Real-world examples: Finding the third side of a triangular plot from two sides and the angle between them.
What should I learn before Law of Cosines in Geometry?
Make sure you are comfortable with Law of Sines in Geometry, the previous topic in Geometry.
What is a common mistake with Law of Cosines in Geometry?
Using the angle that is not between the two known sides.
How can I practise Law of Cosines in Geometry?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Law of Cosines in Geometry?
Move on to Vectors in Geometry next.
Related advanced topics in Geometry
- Heron’s Formula – Triangle area from three sides.
- Regular Polygon Area – Use apothem and perimeter.
- Arc Length of a Circle – Part of the circumference.
- Sector Area – Part of the circle’s area.
- Central and Inscribed Angles – An inscribed angle is half the central angle on the same arc.
- Chord and Tangent Properties – A tangent is perpendicular to the radius at the point of contact.