Formulas & tables/Mathematics/Plane geometry/Area of a triangle (two sides and the included angle)

Area of a triangle (two sides and the included angle)

P=12absin⁡γP=\tfrac12 ab\sin\gamma

Result unit: area units

What it describes

The area of a triangle found from two sides and the angle between them.

Symbols

PP
area
a,ba, b
sides
γ\gamma
angle between the sides

Where it comes from

Area from two sides and the angle between them — useful when you do not know the height. The area is largest when γ=90∘\gamma=90^\circ.

When to use it

You know two sides and the angle between them.

Example

Given a=5a=5, b=8b=8, γ=30∘\gamma=30^\circ. Find the area.
Substitutionsin⁡30∘=0.5\sin30^\circ=0.5.
ResultP=12⋅5⋅8⋅0.5=10P=\tfrac12\cdot5\cdot8\cdot0.5=10.

Related formulas

Area of a triangle (base and height) Law of sines

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