# A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/2, the angle between sides B and C is pi/12, and the length of B is 4, what is the area of the triangle?

Dec 21, 2016

The area of the triangle is:

$S = 8 \cdot \left(2 - \sqrt{3}\right)$

#### Explanation:

This is a right triangle, where $A$ and $B$ are the legs, and $C$ the hypotenuse.

In a right triangle the ratio between the legs equals the tangent of the angle opposite the leg at the numerator, so that we have:

$\frac{A}{B} = \tan \left(\frac{\pi}{12}\right)$

or:

$A = B \tan \left(\frac{\pi}{12}\right) = 4 \left(2 - \sqrt{3}\right)$

And the area is:

$S = \frac{1}{2} A \cdot B = 8 \cdot \left(2 - \sqrt{3}\right)$