How to Find the Area of a Triangle Using Trigonometry

Finding the Area of a 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 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 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.

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See Also

What is a triangle? What is an angle? What is trigonometry? What is the sine function? Find the area of a triangle What is a right triangle? Find the area of a triangle ( interactive widget)