Area of a Triangle (Trigonometric)
Calculate triangle area using A = ½ab·sin(C) when two sides and the included angle are known.
Covers the trig formula alongside Heron and base-height methods.
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
Key Notes
- Formula: Area = ½ab sinC: Given two sides a and b and the included angle C (the angle between them), this formula computes area without needing the height. Equivalent forms: ½bc sinA and ½ac sinB use any two sides with their included angle.
- Connection to base × height: When C = 90°, sinC = 1 and the formula reduces to ½ab — the familiar right-triangle area. For any other angle, sinC scales the area appropriately: maximum area is achieved at C = 90°.
- When to use vs Heron's formula: Use the trig area formula when two sides and an included angle are known (SAS). Use Heron's formula (Area = √(s(s−a)(s−b)(s−c))) when all three sides are known but no angles.
- Negative angles and obtuse triangles: For obtuse triangles (C > 90°), sin C is still positive (sin of an obtuse angle equals sin of its supplement). The formula always gives a positive area regardless of whether C is acute or obtuse.
- Applications: Used in surveying to compute land areas from measured distances and angles, in navigation for triangulation, and in computer graphics to determine the area of triangular mesh faces.