Area of a Triangle (Using Sine)

Area of a Triangle (Using Sine)

How to find the area of a triangle using sine (trigonometric ratio): formula, proof, examples, and their solutions.

Formula

(area) = (1/2)*bc sin A. b, c: Sides. A: Interior angle between the given sides (b, c).

(area) = (1/2)⋅bc sin A

b, c: Sides of a triangle
A: Interior angle between b and c

Proof

Area of a Triangle (Using Sine): Proof of the Formula

Draw the height of the triangle.
And see the left right triangle.

Sine: SOH.
So sin θ = h/c.

So h = c sin A.

The area of the triangle is (1/2)bh.

So (area) = (1/2)bh
= (1/2)bc sin A.

Example 1

Find the area of the given triangle. Sides: 5, 6. Interior angle: 45 degrees.

Sides: 5, 6
Angle: 45º

(area) = (1/2)⋅5⋅6 sin 45º

Draw a 45-45-90 triangle.

The sides of the right triangle are 1, 1, √2.

Sine: SOH.
So sin 45º = 1/√2.

Rationalizing a denominator

Example 2

Find the area of the given triangle. Sides: 4, 7. Interior angle: 120 degrees

Sides: 4, 7
Angle: 120º

(area) = (1/2)⋅4⋅7 sin 120º

To find sin 120º,
draw a right triangle on the coordinate plane
whose reference angle is, 180 - 120, 60º.

So the sides of the right triangle are -1, √3, and 2.

30-60-90 triangle

Sine: SOH.
So sin 120º = √3/2.