Cosine Rule Calculator

Calculation Mode:

Side a:

Side b:

Angle C (degrees):

The Cosine Rule is a fundamental concept in trigonometry that allows you to find unknown sides or angles in any triangle, especially when the triangle is not a right-angled triangle. Whether you're a student solving math problems, an engineer working on design calculations, or just someone curious about geometry, the Cosine Rule Calculator is the perfect tool to make these calculations fast, accurate, and hassle-free.

Our online Cosine Rule Calculator is designed to solve two primary types of problems:

Find a missing side when two sides and the included angle are known (SSS or SAS cases).
Find a missing angle when all three sides of the triangle are known.

This tool provides a step-by-step calculation, displays the formula used, identifies the triangle type, and shows precise results with decimal accuracy.

How to Use the Cosine Rule Calculator

Using the calculator is simple and intuitive. Follow these steps to perform your calculations:

Step 1: Select Calculation Mode

Choose "Find Side (SSS or SAS)" if you want to calculate a missing side using two sides and the angle between them.
Choose "Find Angle (Three Sides Known)" if you have all three sides and want to find an angle.

Step 2: Enter Known Values

For side calculation mode, input the lengths of sides a and b, and the included angle C in degrees.
For angle calculation mode, enter the lengths of all three sides: a, b, and c.

Step 3: Calculate

Click the Calculate button.
The calculator will display the missing side or angle, the formula it used, and the type of triangle (Acute, Right, or Obtuse).
You can reset the calculator anytime using the Reset button.

Example: Using the Cosine Rule Calculator

Let’s walk through an example where you want to find the missing side c in a triangle.

Given:

Side a = 7 units
Side b = 10 units
Angle C = 60°

Process:

Select Find Side (SSS or SAS) mode.
Enter the values 7 for side a, 10 for side b, and 60 for angle C.
Click Calculate.

Result:

The calculator computes side c using the formula:
$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ c2=a2+b2−2ab⋅cos(C)
The calculated side c is approximately 8.66 units.
The triangle type will be shown as Acute Triangle because the angle is less than 90°.

Understanding the Cosine Rule

The Cosine Rule is an extension of the Pythagorean theorem and works for any triangle, not just right-angled ones.

The formula for finding a side is:
$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ c2=a2+b2−2ab⋅cos(C)

Where:

a and b are the known sides
C is the included angle between those sides
c is the side opposite angle C

To find an angle when all three sides are known, the formula is rearranged:
$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$ cos(C)=2aba2+b2−c2

Then use inverse cosine (arccos) to find the angle $C$ C.

Benefits of Using the Cosine Rule Calculator

Accuracy: Avoid manual calculation errors.
Speed: Instantly get results without complex math.
Versatility: Works for all types of triangles.
Educational: Understand the formulas used with detailed outputs.
User-Friendly: Easy to enter values and switch modes.

What Triangle Types Does This Calculator Identify?

The tool automatically classifies the triangle based on the angle C:

Acute Triangle: All angles less than 90°.
Right Triangle: Angle C is exactly 90°.
Obtuse Triangle: Angle C is greater than 90°.

Additional Tips for Using the Calculator

Always input positive numbers for sides and angles.
Angle values must be between 0° and 180° (not inclusive).
Make sure the sum of any two sides is greater than the third side for a valid triangle.
Use the reset button to clear previous entries and start fresh.
Double-check units (all sides should be in the same unit).

15 Frequently Asked Questions (FAQs)

1. What is the Cosine Rule used for?
The Cosine Rule helps find unknown sides or angles in any triangle, especially non-right triangles.

2. Can I use the Cosine Rule for right triangles?
Yes, but the Pythagorean theorem is simpler for right triangles.

3. What inputs do I need to find a missing side?
Two sides and the included angle between them.

4. How do I find an angle with the Cosine Rule?
Input all three sides, and the calculator will find the angle opposite to the side you specify.

5. What if my input values are invalid?
The calculator will alert you if values are invalid or violate triangle inequality.

6. Does the calculator work for obtuse triangles?
Yes, it detects and calculates correctly for obtuse triangles.

7. Why is the angle restricted between 0° and 180°?
Because angles in a triangle must be within this range.

8. Can I use this tool for real-world measurements?
Absolutely, as long as measurements are accurate and consistent.

9. What if I don't know which side is opposite the angle?
Make sure to identify the sides correctly to use the Cosine Rule properly.

10. Can the calculator handle decimal inputs?
Yes, it supports decimal values for precise calculations.

11. Is this tool suitable for students?
Definitely, it's great for students learning trigonometry and geometry.

12. Can I calculate all angles of a triangle?
Yes, by repeating the calculation with different side combinations.

13. How precise are the results?
Results are shown up to four decimal places for accuracy.

14. What if the sum of two sides is less than the third?
This is an invalid triangle, and the calculator will warn you.

15. Can I use this calculator offline?
This version is designed as an online tool, but the logic can be adapted offline.

Conclusion

The Cosine Rule Calculator is an essential and user-friendly online tool for anyone working with triangles. Whether for education, engineering, or hobby projects, this calculator simplifies complex trigonometric calculations into quick, accurate results. By understanding how to input your values and interpret results, you can solve a wide range of triangle problems with confidence.

Try the Cosine Rule Calculator today and make triangle calculations easier than ever!