# A triangle has sides A, B, and C. The angle between sides A and B is (2pi)/3 and the angle between sides B and C is pi/12. If side B has a length of 12, what is the area of the triangle?

Mar 8, 2018

37.5034

#### Explanation:

First, we find the missing angle $\frac{\pi}{12}$ is 15 degrees and $\frac{2 \pi}{3}$ is 120 all triangle's angles add up to 180 180-135 and the missing angle is 45 degrees or $\frac{\pi}{4}$ now that there is a known side, we can use the sin rule.
$\frac{a}{\sin A} = \frac{b}{\sin} B = \frac{c}{\sin} C$
$c = \frac{b \sin C}{\sin} b$

$\frac{12 \sin 120}{\sin} 45$ is 8.188160008 side C
for the next side we get 9.170785943 ide A
http://www.teacherschoice.com.au/Maths_Library/Trigonometry/solve_trig_AAS.htm for more details

the area is 37.5034
here is a good online calculator for this
https://www.mathopenref.com/heronsformula.html
and also how I got to this area.