Answer
$163$ km$^{2}$
Work Step by Step
The area of the triangle is half the product of the length of two sides and the sine of the angle included between them:
$Area=\frac{1}{2}ab \sin C$
We substitute the values of $a,b$ and $C$ in this formula and solve:
$Area=\frac{1}{2}ab \sin C$
$Area=\frac{1}{2}(21.9)(24.6) \sin 142.7^{\circ}$
$Area=\frac{1}{2}(538.74) \sin 142.7^{\circ}$
$Area=269.37 \sin 142.7^{\circ}$
Using a calculator, $\sin 142.7^{\circ}=0.6060$. Therefore,
$Area=269.37 \sin 142.7^{\circ}$
$Area=269.37(0.6060)$
$Area=163.24\approx163$
Therefore, the area of the triangle is $163$ km$^{2}$.
