Triangle Angle Calculator
Calculate missing angles and sides using law of cosines, law of sines, and angle sum theorem
Calculate Triangle Angles
Choose the information you have about the triangle
Triangle Results
All sides must be positive numbers
Example Calculation
3-4-5 Right Triangle
Given: Sides a = 3, b = 4, c = 5
Using Law of Cosines:
• α = arccos((4² + 5² - 3²)/(2×4×5)) = arccos(32/40) ≈ 36.87°
• β = arccos((3² + 5² - 4²)/(2×3×5)) = arccos(18/30) ≈ 53.13°
• γ = 180° - 36.87° - 53.13° = 90°
Result: Right triangle with angles 36.87°, 53.13°, 90°
Triangle Laws
Angle Sum Theorem
α + β + γ = 180°
All triangle angles sum to 180°
Law of Cosines
c² = a² + b² - 2ab·cos(C)
Relates sides and angles
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Ratios of sides to sines
Triangle Tips
Triangle inequality: sum of any two sides > third side
Angles must sum to exactly 180°
Largest angle is opposite the longest side
A triangle can have at most one obtuse angle
Understanding Triangle Angle Calculations
Methods for Finding Angles
1. Three Sides (SSS)
Use the Law of Cosines to find each angle when all three sides are known.
2. Two Sides and Included Angle (SAS)
Find the third side with Law of Cosines, then use Law of Sines for remaining angles.
3. Two Angles
Use the angle sum theorem: the third angle = 180° - sum of other two angles.
Key Theorems
Triangle Angle Sum Theorem
The sum of interior angles in any triangle is always 180°.
Law of Cosines
Generalizes the Pythagorean theorem for any triangle.
Law of Sines
Relates the ratios of sides to the sines of their opposite angles.