Area of a Triangle Using Trigonometry
(KS3, Year 7)
The area of a triangle is found using the formula:

area of a triangle trigonometry formula In this formula, a and b are lengths of two sides of the triangle and C is the angle between them. sin C means finding the sine of the angle C. (sin is the sine function, which is a trigonometric function). The image below shows what we mean by the two sides and the angle between them:

area of a triangle trigonometry You can use other versions of the formula to find the area:

½ bc sin A

½ ca sin B

How to Find the Area of a Triangle Using Trigonometry

Finding the area of a triangle using trigonometry is easy.

Question

What is the area of a triangle with sides of 6 cm and 8 cm with an angle of 30° between them, as shown below?
triangle 6 8 30 degrees

Step-by-Step:

1

Start with the formula:
Area = ½ ab sin C
Don't forget: ½ ab sin C = ½ × a × b × sin C

2

Substitute the length of the sides and the angle between them into the formula. In our example, a = 6, b = 8 and C = 30°.

Area = ½ × 6 × 8 × sin(30°)

Area = ½ × 6 × 8 × 0.5

Area = 12 cm2

Don't forget: ½ × a number = 0.5 × a number = a number ÷ 2.

Answer:

The area of the triangle with with sides of 6 cm and 8 cm with an angle of 30° between them is 12 cm2.

Lesson Slides

The slider below shows another real example of how to find the area of a triangle using trigonometry.

Top Tip

3 Formulas

You can use any two sides and the angle between them to find the area of a triangle. The formulas for the area of the triangle are:
  • ½ ab sin C
  • ½ bc sin A
  • ½ ca sin B
This works as long as the triangle is labelled in a special way (with side a opposite angle A, side b opposite angle B and side c opposite angle C):

labelling triangle

Note

Why Does the Formula Work?

The area of a triangle is given by:
½ × base × height
Use this when you know the length of the base and the height. But what if you only know two sides of the triangle and the angle in between them?

The base is given by b, but what is the height? If we consider the height to be the opposite side of a right triangle with hypotenuse a and angle C:

The height is a sin C.

Area = ½ × base × height

Area = ½ × b × a sin C

Area

= ½ ab sin C

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This page was written by Stephen Clarke.