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Alternate Formula for the Area of a Triangle

Area equals half the product of two sides and the sine of the included angle.

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Sine to find the Area of a Triangle

You have previously learned that the area of a triangle is  where  is a base of the triangle and  is the corresponding height. Why is it helpful to have another formula for calculating the area of a triangle?

Finding the Area of a Triangle by using Sine

One way to find the area of a triangle is by calculating . This formula works when you know or can determine the base and the height of a triangle. What if you wanted to find the area of the following triangle:

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  from  has been drawn. Write an equation that relates  and 7.

Notice that two right triangles have been formed. This means you can use the trigonometric ratios to relate the sides.  is opposite the  angle and 7 is the length of the hypotenuse of the right triangle. This is the sine relationship.

Solving for Unknown Values 

Solve the equation for . Then, use  to find the area of the triangle using your value for . Can you generalize the formula?

If  then . For the height that has been drawn,  is the base.

Notice the calculation: . You found half the product of the two sides and the sine of the included angle. Consider the general triangle below:

If you know , the area will be . If you know , the area will be .

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 . Then, find the area of the triangle.

First find the measure of the exterior angle of the triangle at vertex . Remember that the interior and exterior angles at vertex  must sum to .

Now, consider the right triangle with  as the hypotenuse.  is the side opposite the  angle and 7 is the hypotenuse. Once again, this is the sine relationship.

For the height that has been drawn,  is the base.

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  where  is a base of the triangle and  is the corresponding height. Why is it helpful to have another formula for calculating the area of a triangle?

Often you only know the sides and angles of a triangle and not the height.  allows you to quickly find the area of acute or obtuse triangles when given two sides and an included angle.

Example 2

Find the area of the triangle.

The  angle is the included angle to the two given sides of length 7 and 10. Use the new area formula. 

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  and ). One property of the sine function is that the sine of supplementary angles will always be equal. Therefore, when finding the area of a triangle given an obtuse angle, you can use the obtuse angle in your calculation instead of the acute supplementary angle and your answer will be the same. 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  so that it intersects the extension of  at point . The exterior angle at  is .

Use the new area formula with the  angle.

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 , use .

The result is the same.  and .

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: 

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 .

10. Find the area of the triangle below in two ways. First, use the formula . 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 . Find the measure of  rounded to the nearest degree.

14. Use the triangle below to explain where the area formula  comes from.

15. Why does the area formula  work even if  is obtuse?

Review (Answers)

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

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Vocabulary

Included Angle

The included angle in a triangle is the angle between two known sides.

sine

The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.

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