# If sides A and B of a triangle have lengths of 3 and 12 respectively, and the angle between them is (pi)/6, then what is the area of the triangle?

##### 1 Answer

$A r e a = 9$ square units

#### Explanation:

Use the formula for Area of a triangle when two sides and included angle are given

$A r e a = \frac{1}{2} \cdot a \cdot b \cdot \sin C$
$A r e a = \frac{1}{2} \cdot b \cdot c \cdot \sin A$
$A r e a = \frac{1}{2} \cdot a \cdot c \cdot \sin B$

All of the above are always true.

Given : $a = 3$ and $b = 12$ and $S \in C = S \in \left(\frac{\pi}{6}\right) = \frac{1}{2}$

Use $A r e a = \frac{1}{2} \cdot a \cdot b \cdot \sin C$

$A r e a = \frac{1}{2} \cdot 3 \cdot 12 \cdot \frac{1}{2}$

$A r e a = 9$ square units