You have previously learned that the area of a triangle is

### Finding the Area of a Triangle by using Sine

One way to find the area of a triangle is by calculating

The two given sides are *not* the base and the height. In the examples you will derive a formula for calculating the area of a triangle given this type of information.

#### Writing Equations

Below, the altitude of

Notice that two right triangles have been formed. This means you can use the trigonometric ratios to relate the sides.** **

#### Solving for Unknown Values

Solve the equation for

If

Notice the calculation:

If you know

#### Finding the Area Given Angle Measurements

Previously, you have only used the sine ratio on acute angles. Can you find the area of a triangle when given an obtuse angle? Consider the triangle below. Find an equation to determine the value of

First find the measure of the exterior angle of the triangle at vertex

Now, consider the right triangle with

For the height that has been drawn,

In general, if your given angle is obtuse, you can use the angle supplementary to your given angle in your area calculation.

**Examples**

**Example 1**

Earlier, you were asked why is it helpful to have another formula for calculating the area of a triangle.

You have previously learned that the area of a triangle is

Often you only know the sides and angles of a triangle and not the height.

#### Example 2

Find the area of the triangle.

The

#### Example 3

Find the area of the triangle.

The **sine ratio** is actually a **function** that takes *any* angle measure as an input (not just angles between *Note: You will study the sine function in much more detail in future courses!*

Notice that the given angle is obtuse. Draw the altitude from vertex

Use the new area formula with the

#### Example 4

Find the area of the triangle from #2 using the obtuse angle in your calculations. What do you notice?

Use the area formula as in #3. Instead of using

The result is the same. \begin{align*}\sin 30^\circ=0.5\end{align*} and \begin{align*}\sin 150^\circ=0.5\end{align*}.

**This means that regardless of whether the given angle is acute or obtuse, you can always find the area of a triangle by finding the half the product of two sides and the sine of their included angle:**

**For all triangles: \begin{align*}A=\frac{1}{2} (a)(b) \sin C\end{align*}**

### Review

Find the area of each triangle.

1.

2.

3.

4.

5. Explain why the following triangle has the same area as the triangle in #4.

Find the area of each triangle.

6.

7.

8.

9. Use your calculator to find \begin{align*}\sin 90^\circ\end{align*}.

10. Find the area of the triangle below in two ways. First, use the formula \begin{align*}A=\frac{1}{2} bh\end{align*}. Then, use the new formula from this concept. What do you notice?

11. Find the area of the parallelogram:

12. Use your work from #11 to help you to describe a general method for calculating the area of a parallelogram given its sides and angles.

13. The area of the triangle below is \begin{align*}78 \ un^2\end{align*}. Find the measure of \begin{align*}\theta\end{align*} rounded to the nearest degree.

14. Use the triangle below to explain where the area formula \begin{align*}A=\frac{1}{2} ab \sin C\end{align*} comes from.

15. Why does the area formula \begin{align*}A=\frac{1}{2} ab \sin C\end{align*} work even if \begin{align*}\angle C\end{align*} is obtuse?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.5.