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How to Find the Area of a Triangle Using Trigonometry (Mathematics Lesson)

How to Find the Area of a Triangle Using Trigonometry

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 is the sine function. It is a trigonometric function. sin C means finding the sine of the angle C.

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 (see Note):

A Real Example of How to Find the Area of a Triangle Using Trigonometry

An Example 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 1
Start with the formula:
Area = ½ ab sin C
Don't forget: ½ ab sin C = ½ × a × b × sin (C)

Step 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.

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

Another Real Example of How to Find the Area of a Triangle Using Trigonometry

The slider below shows another real example of how to find the area of a triangle using trigonometry:
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Here's a second test on finding the area of a triangle using trigonometry.
Here's a third test on finding 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):



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