Finding the angle of a triangle given 3 sides is a common task in geometry, engineering, and design. Using the Law of Cosines, you can determine each interior angle from the side lengths alone.
This guide walks through the method step by step, with clear formulas, examples, and practical tips you can apply right away.
| Side a | Side b | Side c | Angle A | Angle B | Angle C |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 36.87° | 53.13° | 90.00° |
| 7 | 10 | 5 | 28.96° | 112.93° | 38.11° |
| 6 | 6 | 6 | 60.00° | 60.00° | 60.00° |
| 8 | 5 | 7 | 81.79° | 38.21° | 60.00° |
Law of Cosines Formula
The Law of Cosines relates the side lengths of a triangle to the cosine of one of its angles. For any triangle with sides a, b, and c opposite angles A, B, and C respectively, the formulas are:
cos(A) = (b² + c² − a²) / (2bc)
cos(B) = (a² + c² − b²) / (2ac)
cos(C) = (a² + b² − c²) / (2ab)
Once you compute the cosine value, use the inverse cosine function (arccos) on your calculator to find the angle in degrees.
Step by Step Calculation
Follow these steps to find the angle of a triangle given 3 sides accurately.
- Label the sides: a, b, and c.
- Choose the angle you want to find, for example angle A.
- Plug the side lengths into the formula: cos(A) = (b² + c² − a²) / (2bc).
- Calculate the numerator and denominator separately.
- Compute the arccos of the result to get angle A in degrees.
- Repeat the process for the other two angles or verify that angles sum to 180°.
Practical Example with Numbers
Consider a triangle with sides a = 7, b = 10, and c = 5. We will find angle A.
First, compute b² + c² − a² = 100 + 25 − 49 = 76. Then compute 2bc = 2 × 10 × 5 = 100. So cos(A) = 76 / 100 = 0.76. Taking arccos(0.76) gives A ≈ 28.96°. You can repeat this for angle B using a² + c² − b² and angle C similarly.
Best Practices and Recommendations
- Label sides and angles clearly before starting calculations.
- Use the Law of Cosines for the first angle to avoid ambiguity.
- Verify your results by checking that angles sum to 180°.
- Use a scientific calculator and set it to degree mode.
- Double-check side lengths and arithmetic to reduce rounding errors.
FAQ
Reader questions
What if the side lengths do not form a valid triangle?
Check the triangle inequality: the sum of any two sides must be greater than the third side. If this fails, no triangle exists and the angle calculation is not possible.
Can I use this method for right triangles only?
No, the Law of Cosines works for any triangle, including right triangles. For right triangles, you can also use the simpler Pythagorean theorem and sine or cosine ratios.
Why does my calculated angle not match the drawing?
This can happen due to rounding errors or mislabeling sides. Ensure side a is opposite angle A, side b opposite angle B, and side c opposite angle C, and use precise values before applying arccos.
How do I find all three angles efficiently?
Calculate one angle using the Law of Cosines, then use the Law of Sines or calculate a second angle similarly. Verify that all three angles add up to 180 degrees to catch mistakes.