﻿ The Side Angle Side Formula to Find A Triangle's Area. Sine to the rescue!

# The Area of a Triangle: SAS Formula

The Side Angle Side Formula

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?

Step 1

Identify two sides and the included angle!

Step 2

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

Step 1

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

Step 2

Apply the formula!

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

Step 1

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

Step 2

Apply the formula!

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

Step 1

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

Step 2

Apply the formula!

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

Step 1

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

Step 2

Apply the formula!

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

Step 1

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

Step 2

Apply the formula!

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

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