ASA Triangle Calculator
Calculate triangle properties using Angle-Side-Angle (ASA) method with step-by-step solutions
ASA Triangle Input
Angle β (Beta)
Side a
Angle γ (Gamma)
Triangle Properties
Complete Measurements
Step-by-Step Solution
Step 1: Find the third angle α α = 180° - β - γ = 180° - 50.0° - 32.0° = 98.0° Step 2: Use Law of Sines to find side b b = a × sin(β) / sin(α) b = 5 × sin(50.0°) / sin(98.0°) b = 5 × 0.7660 / 0.9903 b = 3.8679 cm Step 3: Use Law of Sines to find side c c = a × sin(γ) / sin(α) c = 5 × sin(32.0°) / sin(98.0°) c = 5 × 0.5299 / 0.9903 c = 2.6756 cm Step 4: Calculate area using ASA formula A = (a²/2) × sin(β) × sin(γ) / sin(β + γ) A = (5²/2) × sin(50.0°) × sin(32.0°) / sin(82.0°) A = 12.50 × 0.7660 × 0.5299 / 0.9903 A = 5.1241 cm²
Triangle Validation
Example Calculation
Example: ASA Triangle
Given: β = 50°, a = 5 cm, γ = 32°
Step 1: α = 180° - 50° - 32° = 98°
Step 2: b = 5 × sin(50°) / sin(98°) = 3.868 cm
Step 3: c = 5 × sin(32°) / sin(98°) = 2.676 cm
Step 4: Area = (5²/2) × sin(50°) × sin(32°) / sin(82°) = 5.157 cm²
ASA Triangle Properties
• Two angles and the side between them are known
• The third angle is found using: α = 180° - β - γ
• Remaining sides calculated using Law of Sines
• Area formula: A = (a²/2) × sin(β) × sin(γ) / sin(β + γ)
ASA Triangle Method
Angle-Side-Angle
Two angles and the side between them
Known Values
Angles β, γ and side a
Find Third Angle
α = 180° - β - γ
Law of Sines
Find remaining sides
Key Formulas
Third Angle
α = 180° - β - γ
Law of Sines
a/sin(α) = b/sin(β) = c/sin(γ)
ASA Area
A = (a²/2) × sin(β)sin(γ)/sin(β+γ)
Quick Tips
ASA triangles are always uniquely determined
Sum of angles must equal 180°
All angles must be positive and less than 180°
Used in triangle congruence proofs
Understanding ASA Triangles
What is ASA?
ASA stands for Angle-Side-Angle. This refers to a triangle where two angles and the side between them are known. This configuration allows us to determine all other properties of the triangle uniquely using trigonometric relationships.
Law of Sines
a/sin(α) = b/sin(β) = c/sin(γ)
The Law of Sines states that the ratio of any side of a triangle to the sine of its opposite angle is constant for all three sides and angles.
ASA Triangle Congruence
Two triangles are congruent if they have two angles and the included side equal. This is one of the fundamental congruence postulates in geometry.
Solution Steps
- 1.Find the third angle using angle sum property
- 2.Apply Law of Sines to find unknown sides
- 3.Calculate area using ASA area formula
- 4.Determine triangle type and properties
Applications and Use Cases
Surveying
Land surveyors use ASA triangulation to measure distances and map terrain features.
Navigation
ASA principles help in celestial navigation and GPS triangulation systems.
Engineering
Structural engineers use ASA calculations for truss design and load analysis.
ASA Area Derivation
The ASA area formula is derived from the basic triangle area formula and trigonometry:
Starting with: A = ½ × base × height
Using trigonometry: height = a × sin(γ)
And Law of Sines: base = a × sin(β) / sin(α)
Substituting: A = ½ × (a × sin(β) / sin(α)) × (a × sin(γ))
Simplifying: A = (a²/2) × sin(β) × sin(γ) / sin(α)
Since α = π - β - γ: A = (a²/2) × sin(β) × sin(γ) / sin(β + γ)