# A triangle has sides A,B, and C. If the angle between sides A and B is pi/12, the angle between sides B and C is pi/12, and the length of B is 5, what is the area of the triangle?

Oct 14, 2017

Area of triangle = 3.0026

#### Explanation:

$\angle A = \frac{\pi}{12} , \angle C = \frac{\pi}{12} , \angle B = \frac{5 \pi}{6}$
Side $B = 5$
We know,$\frac{A}{\sin} A = \frac{B}{\sin} B = \frac{C}{\sin} C$
$\frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{5}{\sin} \left(\frac{5 \pi}{6}\right) = \frac{C}{\sin} \left(\frac{\pi}{12}\right)$
$A = \frac{5 \cdot \sin \frac{\pi}{12}}{\sin} \left(\frac{5 \pi}{6}\right) \approx 2.5881$
As $\angle A = \angle C , \angle C \approx 2.5881$

$s = \frac{A + B + C}{2} = \frac{5 + 2.5881 + 2.5881}{2} = 5.0881$
$s - A = s - C = 5.0881 - 2.5881 = 2.5$
$s - B = 5.0881 - 5 = 0.0881$

Area of triangle = $\sqrt{s \cdot \left(s - A\right) \left(s - B\right) \left(s - C\right)}$
$= \sqrt{5.0881 \cdot 2.5881 \cdot 0.0881 \cdot 2.5881} = \approx 3.0026$