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?

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?

Example 1
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

Problem 1
Step 1

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

Step 2

Apply the formula!

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

Problem 2
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!

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

Problem 3
Step 1

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

Step 2

Apply the formula!

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

Problem 4
Step 1

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

Step 2

Apply the formula!

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

Problem 5
Step 1

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

Step 2

Apply the formula!

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