Area of a Triangle using Trigonometry

The area of a triangle refers to the total space or region enclosed by its three sides. Trigonometry provides a powerful method to calculate this area, especially when the height of the triangle is not available, and instead, we have side lengths and angles.

The basic formula for the area of a triangle is:

However, when the height is unknown, trigonometric methods can be used to find the area.

For a triangle with sides a, b and c given as below:

If a and b are two sides of a triangle and C is the included angle between them:

Area =

Similarly,

• Area =
• Area =

Derivation

Consider a triangle with sides a and b, and the angle between them is C. Drop a perpendicular from the vertex opposite to the side C, dividing the triangle into two right triangles.

Using

Area(ABC) = (1/2) × a × h . . . (i)

In triangle ADC, using the sine of angle C,

sin C = h/b

⇒

Now, substitute h = into the equation. (i)

⇒

Thus, the area is:

This formula applies when two sides of the triangle and the angle between them are known.

Solved Examples

Example 1: Find the area of a triangle where a = 7 cm, b = 9 cm, and the angle between them Sin C =45°

• Given: a =7cm , b=9cm and sin c =45°
• Using the formula:

Thus, the area of the triangle is approximately

Example 2: Find the area of a triangle where , , and the angle between them is C

• Given: a = 12 , , and
• Using the formula:

Thus, the area of the triangle is approximately

Example 3: Calculate the area of a triangle with sides a, and the angle between them

• Given: , , and
• Using the formula:

Thus, the area of the triangle is .

Example 4: Find the area of a triangle where , and the angle between them is C

• Given:, and \
• Using the formula:

Thus, the area of the triangle is

Example 5: A triangle has sides a = 9 , , and the angle between them is CFind the area.

• Given: , , and
• Using the formula:

Thus, the area of the triangle is approximately

Practice Problems

Q 1. Find the area of a triangle with a, , and the angle between them C

Q 2. Calculate the area of a triangle where a,, and the angle between them C.

Q 3. A triangle has sides a, , and the angle between them C. Find the area.

Q 4. Determine the area of a triangle with a, , and the angle between them C

Q 5. Find the area of a triangle with sides a, , and the angle between them C

Related Articles

• Heron's Formula
• Area of Isosceles Triangle
• Area of Triangle using Side-Angle-Side
• Area of Triangle with 3 Sides