Triangle Area using Two Sides and Included Angle

\[Area=\frac{1}{2}bc\sin(A)\]

Variables

Areatriangle area (m²)

btriangle side b (m)

ctriangle side c (m)

Aincluded angle between sides b and c (rad or °)

Description

What is this formula?

This trigonometric formula calculates the area of a triangle using two known sides and the included angle between them.

Unlike the standard base-height area formula, this method is especially useful when the height of the triangle is unknown but angular information is available.

It is one of the most common area formulas used in trigonometry and geometry.

When to use it

Use this formula when:

- Two sides of a triangle are known

- The angle between those sides is known

- The triangle height is not directly available

- Solving oblique triangle problems

- Working with surveying or navigation calculations

The formula works for any triangle type.

Example

If:

b = 8 m

c = 10 m

A = 35°

Then:

Area = (8*10*sin(35°))/2

Area ≈ 22.94 m²

Applications

- Trigonometry and geometry

- Civil engineering

- Land surveying

- Navigation calculations

- Physics and mechanics

- Architecture and design