# A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, respectively. The angle between A and C is (11pi)/24 and the angle between B and C is  (11pi)/24. What is the area of the triangle?

Jan 15, 2016

First of all let me denote the sides with small letters $a$, $b$ and $c$.
Let me name the angle between side $a$ and $b$ by $\angle C$, angle between side $b$ and $c$ by $\angle A$ and angle between side $c$ and $a$ by $\angle B$.

Note:- the sign $\angle$ is read as "angle".
We are given with $\angle B$ and $\angle A$. We can calculate $\angle C$ by using the fact that the sum of any triangles' interior angels is $\pi$ radian.
$\implies \angle A + \angle B + \angle C = \pi$
$\implies \frac{11 \pi}{24} + \frac{11 \pi}{24} + \angle C = \pi$
$\implies \angle C = \pi - \left(\frac{11 \pi}{24} + \frac{11 \pi}{24}\right) = \pi - \frac{11 \pi}{12} = \frac{\pi}{12}$
$\implies \angle C = \frac{\pi}{12}$

It is given that side $a = 7$ and side $b = 2.$

Area is also given by
$A r e a = \frac{1}{2} a \cdot b S \in \angle C$

$\implies A r e a = \frac{1}{2} \cdot 7 \cdot 2 S \in \left(\frac{\pi}{12}\right) = 7 \cdot 0.2588 = 1.8116$ square units
$\implies A r e a = 1.8116$ square units