Area of a Triangle SAS Calculator
Calculate triangle area using two sides and the included angle (Side-Angle-Side method)
Calculate Triangle Area (SAS Method)
Length of the first side
Length of the second side
Angle between the two sides
Triangle Properties
Example Calculation
Example 1: Right Triangle
Given: Side a = 3 cm, Side b = 4 cm, Angle γ = 90°
Formula: Area = ½ × a × b × sin(γ)
Calculation: Area = ½ × 3 × 4 × sin(90°) = ½ × 3 × 4 × 1 = 6 cm²
Example 2: Acute Triangle
Given: Side a = 5 cm, Side b = 7 cm, Angle γ = 60°
Calculation: Area = ½ × 5 × 7 × sin(60°) = ½ × 5 × 7 × 0.866 = 15.16 cm²
SAS Method
Side-Angle-Side
Two sides and included angle
Formula
Area = ½ × a × b × sin(γ)
Trigonometry
Uses sine function
Valid Range
Angle: 0° < γ < 180°
Triangle Types
Acute Triangle
All angles < 90°
Right Triangle
One angle = 90°
Obtuse Triangle
One angle > 90°
Quick Tips
The angle must be between the two given sides
Maximum area occurs when angle = 90°
sin(180° - θ) = sin(θ)
Use law of cosines to find third side
Understanding the SAS Triangle Area Formula
What is the SAS Method?
The SAS (Side-Angle-Side) method allows you to calculate the area of a triangle when you know two sides and the angle between them (the included angle). This is one of the most practical methods for finding triangle area in real-world applications.
The Formula
Area = ½ × a × b × sin(γ)
- a, b: The two known sides
- γ: The included angle between the sides
- sin(γ): The sine of the included angle
How It Works
The formula is derived from the basic area formula (½ × base × height) using trigonometry. If we consider one side as the base, the height can be calculated using the sine function and the other known side.
Step-by-Step Process
- 1.Identify the two sides and the included angle
- 2.Convert angle to radians if necessary
- 3.Calculate sin(γ)
- 4.Apply the formula: ½ × a × b × sin(γ)
Applications and Use Cases
Engineering
Calculating areas of triangular structures, roof sections, and mechanical components.
Architecture
Determining floor areas, triangular sections in building design and landscaping.
Navigation
Calculating distances and areas in GPS mapping and surveying applications.