Area of a Triangle (Trigonometric)
Calculate the area of any triangle using A = ½ab·sin(C).
Find the area when you know two sides and the included angle.
The Formula
A = ½ × a × b × sin(C)
This formula calculates the area of any triangle when you know two sides and the angle between them.
It works for all triangles — not just right triangles.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the triangle |
| a, b | Two known sides of the triangle |
| C | The angle between sides a and b (the included angle) |
| sin | The sine function |
Example 1
Find the area of a triangle with sides 8 and 12, and an included angle of 30°
a = 8, b = 12, C = 30°
A = ½ × 8 × 12 × sin(30°)
A = ½ × 96 × 0.5
A = 24 square units
Example 2
Two sides of a triangular field are 50 m and 70 m with an angle of 65° between them. Find the area.
a = 50, b = 70, C = 65°
A = ½ × 50 × 70 × sin(65°)
A = ½ × 3,500 × 0.9063
A = ½ × 3,172.05
A ≈ 1,586.03 m²
When to Use It
Use this formula when:
- You know two sides and the angle between them (SAS)
- The perpendicular height is not known or hard to measure
- Working with surveying, navigation, or land measurement
- When you only know side lengths (no angle), use Heron's formula instead