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# Derivation of the Triangle Area Formula

## Derive and apply area equals half the product of two sides and the sine of the included angle.

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Practice Derivation of the Triangle Area Formula
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Area of a Triangle

From geometry you already know that the area of a triangle is .

What if you are given the sides of a triangle are 5 and 6 and the angle between the sides is ? You are not directly given the height, but can you still figure out the area of the triangle?

#### Watch This

http://www.youtube.com/watch?v=mBFDq4bPXMs James Sousa: The Area of a Triangle Using Sine

#### Guidance

The sine function allows you to find the height of any triangle and substitute that value into the familiar triangle area formula.

Using the sine function, you can isolate  for height:

Substituting into the area formula:

Example A

Given  with . What is the area?

Solution: The letters don’t have to match exactly because the triangle or the formula can just be relabeled. The important part is that neither given side corresponds to the given angle.

Example B

Given  has area 28 square inches, what is the angle included between side length 8 and 9?

Solution:

Example C

Given triangle  with  and , what is the length of side

Solution:

Concept Problem Revisited

What if you are given the sides of a triangle are 5 and 6 and the angle between the sides is

#### Vocabulary

The included angle between two sides of a triangle is the angle at the point where the two sides meet.

#### Guided Practice

1. What is the area of  with ?

2. What is the area of  with ?

3. The area of a triangle is 3 square units. Two sides of the triangle are 4 units and 5 units. What is the measure of their included angle?

1.

2. Because none of the angles are given, there are two possible solution paths. You could use the Law of Cosines to find one angle. The angle opposite the side of length 11 is  therefore the area is:

Another way to find the area is through the use of Heron’s Formula which is:

Where  is the semi perimeter:

3.

#### Practice

For 1-11, find the area of each triangle.

1.  if , and .
2.  if , and .
3.  if , and .
4.  if , and .
5.  if , and .
6.  if , and .
7.  if , and .
8.  if , and .
9.  if , and .
10.  if , and .
11.  if , and .
12. The area of a triangle is 12 square units. Two sides of the triangle are 8 units and 4 units. What is the measure of their included angle?
13. The area of a triangle is 23 square units. Two sides of the triangle are 14 units and 5 units. What is the measure of their included angle?
14. Given  has area 32 square inches, what is the angle included between side length 9 and 10?
15. Given  has area 15 square inches, what is the angle included between side length 7 and 11?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.7.