SAS Triangle Calculator
Calculate all properties of a triangle using two sides and the included angle (Side-Angle-Side)
Calculate SAS Triangle
First adjacent side
Second adjacent side
Included angle between sides a and b
Triangle Properties
Enter two sides and the included angle to calculate triangle properties
Example Calculation
Example: Find triangle with sides 4 cm, 3 cm and angle 42°
Given: a = 4 cm, b = 3 cm, γ = 42°
Step 1: c = √(4² + 3² - 2×4×3×cos(42°)) = √(16 + 9 - 24×0.7431) = √7.1656 ≈ 2.68 cm
Step 2: α = arcsin(sin(42°) × 4 ÷ 2.68) = arcsin(0.9994) ≈ 89.41°
Step 3: β = 180° - 89.41° - 42° = 48.59°
Step 4: Area = ½ × 4 × 3 × sin(42°) = 6 × 0.6691 ≈ 4.01 cm²
Step 5: Perimeter = 4 + 3 + 2.68 = 9.68 cm
Triangle Laws Used
Law of Cosines
c² = a² + b² - 2ab·cos(γ)
Used to find the third side
Law of Sines
a/sin(α) = b/sin(β) = c/sin(γ)
Used to find remaining angles
SAS Area Formula
A = ½ab·sin(γ)
Direct area calculation from SAS
Triangle Classification
By Angles:
- • Acute: All angles < 90°
- • Right: One angle = 90°
- • Obtuse: One angle > 90°
By Sides:
- • Scalene: All sides different
- • Isosceles: Two sides equal
- • Equilateral: All sides equal
Understanding SAS Triangles
What is a SAS Triangle?
A SAS (Side-Angle-Side) triangle is defined by two adjacent sides and the included angle between them. This combination uniquely determines all other properties of the triangle, making SAS triangles congruent.
Why SAS Triangles are Special
- •Uniquely defined - no ambiguity in construction
- •Congruent triangles - same SAS values create identical triangles
- •Can calculate all other properties using trigonometry
- •Foundation for many geometric proofs and applications
Solution Method
Step 1: Find Third Side
Use Law of Cosines: c² = a² + b² - 2ab·cos(γ)
Step 2: Find First Unknown Angle
Use Law of Sines: sin(α)/a = sin(γ)/c
Step 3: Find Last Angle
Use angle sum: α + β + γ = 180°
Step 4: Calculate Area & Perimeter
Area: A = ½ab·sin(γ), Perimeter: P = a + b + c
SAS vs Other Triangle Types
SAS (Side-Angle-Side)
Two sides + included angle
✓ Unique solution
SSA (Side-Side-Angle)
Two sides + non-included angle
⚠️ May have 0, 1, or 2 solutions
SSS (Side-Side-Side)
Three sides given
✓ Unique solution