How to Calculate the Angles of a Triangle
Every triangle has three interior angles that always sum to exactly 180°. Knowing this, plus the relationships between sides and angles, lets you solve for unknown angles in any triangle.
The Basic Rule
Angle A + Angle B + Angle C = 180°
If you know two angles, the third is always:
Angle C = 180° − Angle A − Angle B
Finding Angles Using the Law of Cosines
When you know all three sides (SSS), use the Law of Cosines:
cos(A) = (b² + c² − a²) / (2bc)
Where a, b, c are the side lengths opposite to angles A, B, C respectively.
Step-by-Step Example (SSS)
A triangle has sides a = 7, b = 5, c = 8. Find angle A.
- Apply Law of Cosines: cos(A) = (5² + 8² − 7²) / (2 × 5 × 8)
- Calculate numerator: 25 + 64 − 49 = 40
- Calculate denominator: 80
- cos(A) = 40/80 = 0.5
- A = arccos(0.5) = 60°
Finding Angles Using the Law of Sines
When you know one angle and its opposite side:
sin(A)/a = sin(B)/b = sin(C)/c
Right Triangle Special Case
In a right triangle (one 90° angle), you can use basic trigonometry:
tan(θ) = opposite / adjacent
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
Practical Applications
- Construction: Calculating roof angles and rafter cuts
- Navigation: Triangulation to determine position
- Physics: Resolving force vectors into components
Use our triangle calculator to find all angles from any combination of sides and angles.