The **Side Angle Side formula** for finding the area of a triangle is a way to use the sine trigonometric function to calculate the height of a triangle and use that value to find the area of the triangle

### The formula

$$
\text{Area } =\frac{1}{2} \cdot c \cdot b \cdot sin(\text{A})
$$

or, in general

$$
Area = \frac{1}{2} \cdot side_1 \cdot side_2 \cdot sin(\text{included angle})
$$

Visit this url, if you want to review what is meant by 'included angle'.

#### Where does this formula come from?

**Answer**

We all know that the general formula for the area of a triangle is $$A= \frac{1}{2} \cdot base \cdot height $$

Well, look at the picture below, the question is, *how do we get the height of the triangle?*

Well, we can use sine to solve for the side length

$$
sin(68) = \frac{h}{8}
$$

$$
h = 8 \cdot sin(68)
$$

#### Can you identify which triangle below has an included angle?

Identify two sides and the included angle!

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 8 \cdot 12 \cdot sin(48) \\ = 3.567 $$

**Practice** Problems

We cannot use the 54 angle because we need the **included** angle.

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 165 \cdot 131 \cdot sin(79) \\ = 10,608.9 $$

We cannot use the 190 side length because we need the sides that include the only angle that we know!.

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 145 \cdot 170 \cdot sin(67) \\ = 11411.96 $$

We cannot use the 22 ° angle, because it is not the included angle.

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 7 \cdot 9 \cdot sin(115) \\ = 28.549 $$

We cannot use the 33° or the 26° angle, because they are not included angles.

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 4 \cdot 5 \cdot sin(121) \\ = 8.57 $$

We cannot use the 41° or the 28° angle, because they are not included angles.

Apply the formula!

$$ A = \frac{1}{2} \cdot c\cdot b\cdot sin(A) \\ A = \frac{1}{2} \cdot 10 \cdot 14 \cdot sin(110) \\ = 65.8 $$