The area of a triangle can be found using trigonometry when two sides and the included angle are known. This method works for any triangle, not just right angled triangles.

 

The Trigonometric Area Formula

The formula for the area of a triangle is:

$$
area = \frac{1}{2}ab\sin C
$$

 

In this formula:
• \( a \) and \( b \) are the lengths of two sides
• \( C \) is the angle between those two sides

 

The angle must be the included angle, meaning it lies between the two known sides.

 

Using the wrong angle will give an incorrect answer

 

 

When to Use This Formula

This formula is used when:
• two sides of a triangle are known
• the angle between those sides is known
• the triangle is not right angled

 

It is especially useful when the perpendicular height is difficult to find.

 

 

Applying the Formula

To find the area:
• identify the two known sides
• identify the angle between them
• substitute values into the formula
• calculate using a calculator

 

Example Structure

If sides are \( a = 8 \) cm and \( b = 5 \) cm, and the included angle is \( 40^\circ \):

$$
area = \frac{1}{2} \times 8 \times 5 \times \sin 40^\circ
$$

 

The answer will be given in square units, such as square centimetres.

 

 

Units and Accuracy

The area is always measured in square units.

 

If side lengths are in centimetres, the area will be in square centimetres.

 

Angles must be entered in degrees unless stated otherwise.

 

Check your calculator is in degree mode

 

 

Key Points to Remember

The formula is \( \frac{1}{2}ab\sin C \).
The angle must be between the two known sides.
This formula works for any triangle.
The answer is always in square units.
Correct identification of the included angle is essential.

 

Using the trigonometric area formula allows the area of non right angled triangles to be calculated accurately and efficiently.