AAS Triangle Calculator
Calculate triangle dimensions from two angles and one non-included side
AAS Triangle Calculator
The side opposite to angle α
Angle opposite to side a
Angle adjacent to side a
Triangle Results
Enter valid triangle dimensions to see results
Make sure angles are positive and sum to 180°
Example Calculations
Standard AAS Triangle
Given: Side a = 16 cm, Angle α = 40°, Angle β = 25°
Calculate: γ = 180° - 40° - 25° = 115°
Side b: b = 16 × sin(25°) / sin(40°) ≈ 10.53 cm
Side c: c = 16 × sin(115°) / sin(40°) ≈ 22.58 cm
Area: A = (16²/2) × sin(25°) × sin(65°) / sin(40°) ≈ 76.27 cm²
Right Triangle AAS
Given: Side a = 10 cm, Angle α = 90°, Angle β = 30°
Calculate: γ = 180° - 90° - 30° = 60°
Result: This forms a 30-60-90 right triangle
Area: A ≈ 43.30 cm²
Obtuse Triangle AAS
Given: Side a = 8 cm, Angle α = 120°, Angle β = 35°
Calculate: γ = 180° - 120° - 35° = 25°
Result: Obtuse triangle (one angle > 90°)
Area: A ≈ 13.58 cm²
AAS Triangle Properties
Angle-Angle-Side
Two consecutive angles and the non-included side
Known: α, β, and side a
Law of Sines
a/sin(α) = b/sin(β) = c/sin(γ)
Core formula for calculations
Congruence
AAS triangles are congruent if dimensions match
Same shape and size
Quick Reference
Third angle: γ = 180° - α - β
Missing sides calculated using Law of Sines
Area formula: A = (a²/2) × sin(β) × sin(α+β) / sin(α)
Height: h = a × sin(γ)
Understanding AAS Triangles
What is an AAS Triangle?
An AAS (Angle-Angle-Side) triangle is defined when you know two angles and the side that is NOT between them (non-included side). This is different from ASA where the side is between the two known angles.
AAS Triangle Congruence
Two triangles are congruent by AAS if two consecutive angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle.
Key Formulas
Third angle: γ = 180° - α - β
Law of Sines: a/sin(α) = b/sin(β) = c/sin(γ)
Area: A = (a²/2) × sin(β) × sin(α+β) / sin(α)
Step-by-Step Solution Process
Find the third angle
Use the fact that angles sum to 180°
Apply Law of Sines
Calculate missing sides b and c
Calculate area and height
Use trigonometric area formula
Applications
- •Surveying and navigation
- •Engineering design and construction
- •Physics problems involving vectors
- •Computer graphics and 3D modeling