Triangle Side Angle Calculator
Calculate missing sides and angles in triangles using Law of Cosines, Law of Sines, and trigonometry
Calculate Triangle Sides and Angles
Results - Three Sides (SSS)
Example Calculations
Example 1: Three Sides (SSS)
Given: Sides = 4, 5, 6 units
Law of Cosines: cos(A) = (b² + c² - a²) / (2bc)
Solution: cos(A) = (5² + 6² - 4²) / (2×5×6) = 45/60 = 0.75
Result: Angle A = arccos(0.75) ≈ 41.41°
All angles: A ≈ 41.41°, B ≈ 55.77°, C ≈ 82.82°
Example 2: Two Sides + Included Angle (SAS)
Given: Sides a = 8, b = 6, included angle C = 60°
Law of Cosines: c² = a² + b² - 2ab cos(C)
Solution: c² = 8² + 6² - 2(8)(6)cos(60°) = 64 + 36 - 48 = 52
Result: Side c = √52 ≈ 7.21 units
Other angles: A ≈ 73.90°, B ≈ 46.10°
Example 3: Two Angles + One Side (AAS)
Given: Angles A = 30°, B = 45°, side opposite to A = 5
Third angle: C = 180° - 30° - 45° = 105°
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Solution: b = a × sin(B)/sin(A) = 5 × sin(45°)/sin(30°) ≈ 7.07
Result: All sides ≈ 5, 7.07, 9.66 units
Triangle Solving Methods
Three Sides
Use Law of Cosines to find all angles
Two Sides + Angle
Law of Cosines then Law of Sines
Two Angles + Side
Use Law of Sines for all sides
Key Formulas
Law of Cosines
Law of Sines
Angle Sum
Quick Tips
Always check triangle inequality: sum of any two sides > third side
Use Law of Cosines when you know SAS or SSS
Use Law of Sines when you know AAS, ASA, or SSA
All angles should sum to exactly 180°
Understanding Triangle Side and Angle Calculations
Triangle Congruence Cases
SSS (Side-Side-Side)
When you know all three sides, use the Law of Cosines to find all angles.
- Most straightforward method
- Always produces a unique triangle
- Check triangle inequality first
SAS (Side-Angle-Side)
When you know two sides and the included angle between them.
- Use Law of Cosines to find third side
- Then use Law of Sines for remaining angles
- Always produces a unique triangle
AAS/ASA (Angle-Angle-Side)
When you know two angles and one side.
- Find third angle using angle sum property
- Use Law of Sines to find remaining sides
- Always produces a unique triangle
Mathematical Foundations
Law of Cosines
Generalizes the Pythagorean theorem for any triangle:
c² = a² + b² - 2ab cos(C)
Where C is the angle opposite to side c.
Law of Sines
Relates sides and angles through sine ratios:
a/sin(A) = b/sin(B) = c/sin(C)
This ratio equals the diameter of the circumcircle.
Applications
- • Surveying and navigation
- • Engineering and construction
- • Computer graphics and game development
- • Physics and astronomy
- • Architecture and design