Angle Calculator Using Cosine Rule

How To Calculate An Angle Knowing 2 Sides

Cosine Rule Formula:

\[ \theta = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \]

Angle θ:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Cosine Rule?

The cosine rule (also known as the law of cosines) relates the lengths of the sides of a triangle to the cosine of one of its angles. It's particularly useful for finding:

An angle when you know all three sides of a triangle
The third side when you know two sides and the included angle

2. How Does the Calculator Work?

The calculator uses the cosine rule formula:

\[ \theta = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \]

Where:

\( a \) — Length of side a
\( b \) — Length of side b
\( c \) — Length of side opposite the angle θ
\( \theta \) — The angle between sides a and b (in degrees)

Explanation: The formula calculates the cosine of the angle using the three side lengths, then takes the inverse cosine (arccos) to find the angle in radians, which is then converted to degrees.

3. When to Use the Cosine Rule

Details: Use the cosine rule when:

You know all three sides of a triangle and need to find an angle
You know two sides and the included angle and need to find the third side

It works for any type of triangle, not just right-angled triangles.

4. Using the Calculator

Tips:

Enter all three side lengths in the same units
Side c should be the side opposite the angle you want to find
All side lengths must be positive numbers
The sum of any two sides must be greater than the third side (triangle inequality)

5. Frequently Asked Questions (FAQ)

Q1: What if I get an error or no result?
A: This usually means the side lengths don't form a valid triangle. Check that the sum of any two sides is greater than the third side.

Q2: Can I use this for right-angled triangles?
A: Yes, but for right-angled triangles, using the Pythagorean theorem and basic trigonometry might be simpler.

Q3: How accurate is the calculation?
A: The calculation is mathematically exact, but real-world measurements may have precision limitations.

Q4: What units should I use?
A: Any consistent units (cm, m, inches, etc.) as long as all three sides are in the same units.

Q5: Can I calculate a side length instead of an angle?
A: Yes, the cosine rule can be rearranged to find a side length when you know two sides and the included angle.