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# Area of a Triangle Using Trigonometry

(KS3, Year 7)

## The Lesson

The area of a triangle is found using the 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:

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?## 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 cm

^{2}**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 cm^{2}.

## 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

## 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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