# Area of Triangle in Terms of Two Sides and Angle

## Corollary to Area of Triangle in Terms of Side and Altitude

The area of a triangle $ABC$ is given by:

$\dfrac 1 2 a b \sin C$

where:

$a, b$ are two of the sides
$C$ is the angle of the vertex opposite its other side $c$.

## Proof 1

 $\displaystyle \map \Area {ABC}$ $=$ $\displaystyle \frac 1 2 h c$ Area of Triangle in Terms of Side and Altitude $\displaystyle$ $=$ $\displaystyle \frac 1 2 h \paren {p + q}$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 a b \paren {\frac p a \frac h b + \frac h a \frac q b}$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 a b \paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta}$ Definition of Sine of Angle and Definition of Cosine of Angle $\displaystyle$ $=$ $\displaystyle \frac 1 2 a b \, \map \sin {\alpha + \beta}$ Sine of Sum $\displaystyle$ $=$ $\displaystyle \frac 1 2 a b \sin C$

$\blacksquare$

## Proof 2

By definition of sine:

$h = b \sin C$
$\map \Area {ABC} = \dfrac {a h} 2$

Substituting:

$\map \Area {ABC} = \dfrac {a b \sin C} 2$

$\blacksquare$