# A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/8, the angle between sides B and C is pi/6, and the length of B is 13, what is the area of the triangle?

Apr 23, 2018

$\textrm{a r e a} \approx 20.3798$

#### Explanation:

First of all, let's rewrite in standard notation. It really helps to do these if we always label the triangle consistently.

The triangle has sides $a , b , c$ (small letters) where $b = 13$ with opposing angles $C = \frac{\pi}{8} = {22.5}^{\circ}$, $A = \frac{\pi}{6} = {30}^{\circ}$.

The remaining angle is

$B = \pi - A - C = \pi - \frac{\pi}{8} - \frac{\pi}{6} = \frac{17 \pi}{24} = {127.5}^{\circ}$

The Law of Sines says

$\frac{b}{\sin} B = \frac{c}{\sin} C$

$c = \frac{b \sin C}{\sin} B = \frac{13 \sin \left({22.5}^{\circ}\right)}{\sin} \left({127.5}^{\circ}\right)$

Now the area is

$\textrm{a r e a} = \frac{1}{2} b c \sin A = \frac{{\left(13\right)}^{2} \sin \left({30}^{\circ}\right) \sin \left({22.5}^{\circ}\right)}{2 \setminus \sin {127.5}^{\circ}}$

I'm tempted to work out an exact answer, because those sines are all expressible using the usual operations including square root. But it's late, and the calculator says

$\textrm{a r e a} \approx 20.3798$

Jun 18, 2018

color(maroon)(A_t == 20.37 " sq units"

#### Explanation:

$b = 13 , \hat{A} = \frac{\pi}{6} , \hat{C} = \frac{\pi}{8} , \hat{B} = \pi - \frac{\pi}{6} - \frac{\pi}{8} = \frac{17 \pi}{24}$

Applying the Law of sines,

$a = \frac{b \sin A}{\sin} B = \frac{13 \cdot \sin \left(\frac{\pi}{6}\right)}{\sin} \left(\frac{17 \pi}{24}\right) \approx 8.19$

Formula for area of " Delta, A_t = (1/2) * a b * sin C

color(maroon)(A_t = (1/2) * 8.19 * 13 * sin ( pi/8) = 20.37 " sq units"#