Cosine Rule Calculator
Calculate unknown triangle sides and angles with the cosine rule formula
Triangle Calculator
Results
Enter your triangle measurements and click calculate to see the results.
About the Cosine Rule
What is the Cosine Rule?
The cosine rule (also called the law of cosines) is a fundamental theorem in trigonometry that relates the lengths of the sides of any triangle to the cosine of one of its angles. Unlike the Pythagorean theorem, which only applies to right triangles, the cosine rule works for all triangles.
This formula allows you to calculate the third side of a triangle when you know two sides and the included angle, or find any angle when you know all three sides.
When to Use the Cosine Rule
- Finding the third side when you know two sides and the included angle (SAS)
- Finding any angle when you know all three sides (SSS)
- Solving non-right triangles in engineering and surveying
- Navigation problems involving distances and bearings
- Physics problems involving vectors and forces
Step-by-Step Method
- Identify what you know and what you need to find
- Choose the appropriate form of the cosine rule
- Substitute the known values into the formula
- Solve the equation carefully, paying attention to units
- Check your answer makes sense (triangle inequality)
- Round to appropriate significant figures
Alternative Forms
The cosine rule can be rearranged to find different unknowns:
Real-World Applications
Architecture: Calculating roof angles and structural supports in non-rectangular buildings.
Navigation: Determining distances between points when direct measurement isn’t possible.
Engineering: Analysing forces in mechanical systems and bridge design.
Astronomy: Calculating distances between celestial objects using triangulation.
Surveying: Measuring land boundaries and creating accurate maps.
Tips for Success
Check triangle validity: Ensure the sum of any two sides is greater than the third side.
Angle units: Make sure your calculator is in the correct mode (degrees or radians).
Rounding: Keep full precision during calculations and round only the final answer.
Verification: Use the fact that angles in a triangle sum to 180 degrees to check your work.
Frequently Asked Questions
The cosine rule is used when you know either two sides and the included angle (SAS) or all three sides (SSS). The sine rule is used when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). The cosine rule is generally more reliable as it doesn’t have the ambiguous case that can occur with the sine rule.
The cosine rule is actually a generalisation of Pythagoras’ theorem. When the angle C is 90 degrees, cos(90°) = 0, so the formula c² = a² + b² – 2ab cos(C) becomes c² = a² + b², which is Pythagoras’ theorem. This means the cosine rule works for all triangles, including right triangles.
Yes, the cosine rule applies to all triangles – acute, obtuse, and right triangles. This universal applicability makes it one of the most important tools in trigonometry. However, you must ensure you have the correct combination of known values: either two sides and the included angle, or all three sides.
If you get a negative value under the square root when calculating a side length, it means the triangle doesn’t exist with the given measurements. Check that your input values satisfy the triangle inequality: the sum of any two sides must be greater than the third side. Also verify that angles are between 0° and 180°.
This calculator provides results accurate to several decimal places, suitable for most practical applications. However, remember that the accuracy of your final answer depends on the precision of your input measurements. For professional engineering or surveying work, consider the appropriate number of significant figures for your specific application.