A triangle has sides A, B, and C. The angle between sides A and B is (pi)/6 and the angle between sides B and C is pi/6. If side B has a length of 3, what is the area of the triangle?

Dec 6, 2016

${\text{Area}}_{\triangle} = \textcolor{g r e e n}{\frac{9 \sqrt{3}}{4}}$

Explanation:

As can be seen from the image below, the given triangle can be split into 2 triangles with angles: $\frac{\pi}{6} , \frac{\pi}{3} , \mathmr{and} \frac{\pi}{2}$
This is the ratios for one of that standard trigonometric triangles
we can see that the height of the original triangle must be $\textcolor{red}{\frac{3}{2} \cdot \sqrt{3}}$

and the triangle's area will be:
$\textcolor{w h i t e}{\text{XXX}} \frac{1}{2} \cdot b \cdot h = \frac{1}{2} \times 3 \times \left(\frac{3 \sqrt{3}}{2}\right) = \frac{9 \sqrt{3}}{4}$