# A triangle has sides A, B, and C. Sides A and B have lengths of 5 and 3, respectively. The angle between A and C is (17pi)/24 and the angle between B and C is  (5pi)/24. What is the area of the triangle?

Apr 9, 2017

$\approx 1.94$

#### Explanation:

The area of a triangle can be calculated using the formula $A = \frac{1}{2} a b \sin \left(\gamma\right)$, where $a$ and $b$ are two of the side lengths of the triangle and $\gamma$ is the angle between the two sides.

We are given two of the side lengths, $5$ and $3$. The measure of the angle between them is not given but can be calculated using the measures of the two other angles given and the fact that the sum of the angles in a triangle is $\pi$.

The angle between $A$ and $B$ is $\pi - \frac{17 \pi}{24} - \frac{5 \pi}{24} = \frac{\pi}{12}$.

Substitute these values into the formula $A = \frac{1}{2} a b \sin \left(\gamma\right) = \frac{1}{2} \cdot 5 \cdot 3 \cdot \sin \left(\frac{\pi}{12}\right) \approx 1.94$.