A triangle has sides A, B, and C. Sides A and B have lengths of 8 and 10, respectively. The angle between A and C is (5pi)/8 and the angle between B and C is  (5pi)/24. What is the area of the triangle?

Mar 18, 2018

The area is $20$ square units by the calculation below.

Explanation:

The area is half the product of two sides times the sine of the included angle between them. Here you have two sides but the other two angles. You need the third angle which is the one you want in the formula.

Use the fact that the angles of a triangle add up to 180° or $\setminus \pi$ radians. So

$\setminus \frac{5 \setminus \pi}{8} + \setminus \frac{5 \setminus \pi}{24} + \setminus \theta = \setminus \pi$

$\setminus \frac{5 \setminus \pi}{6} + \setminus \theta = \setminus \pi$

Then $\setminus \theta = \setminus \frac{\setminus \pi}{6}$, and this is the included angle between A and B we require. Therefore:

$A r e a = \left(\setminus \frac{1}{2}\right) \left(8\right) \left(10\right) \setminus \sin \left(\setminus \frac{\setminus \pi}{6}\right) = \left(\setminus \frac{1}{2}\right) \left(8\right) \left(10\right) \left(\setminus \frac{1}{2}\right) = 20$