# A triangle has sides A, B, and C. The angle between sides A and B is (7pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 8, what is the area of the triangle?

Oct 20, 2017

Area of $\Delta A B C = \ast 196.0188 \ast$

#### Explanation:

Three angles are $\left(\frac{7 \pi}{12} , \frac{\pi}{12} , \frac{\pi}{3}\right)$

$\frac{C}{\sin} \left(\angle C\right) = \frac{A}{\sin} \left(\angle A\right) = \frac{B}{\sin} \left(\angle B\right)$
$\frac{C}{\sin} \left(\frac{7 \pi}{12}\right) = \frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{8}{\sin} \left(\frac{\pi}{3}\right)$

$C = \frac{8 \cdot \sin \left(\frac{7 \pi}{12}\right)}{\sin} \left(\frac{\pi}{3}\right) = 8.9228$

$A = \frac{8 \cdot \sin \left(\frac{\pi}{12}\right)}{\sin} \left(\frac{\pi}{3}\right) = 2.3909$

Semi Perimeter of $\Delta A B C s = \frac{8 + 8.9228 + 2.3909}{2}$
$s = 19.3137$
$s - a = 19.3137 - 2.3909 = 16.9228$
$s - b = 19.3137 - 8 = 11.3137$
$s - c = 19.3137 - 8.9228 = 10.3909$

Area of $\Delta A B C = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

Area of $\Delta A B C = \sqrt{19.3137 \cdot 16.9228 \cdot 11.3137 \cdot 10.3909}$
= 196.0188