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# Derivation of the Triangle Area Formula

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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 $\frac{1}{2} \cdot b \cdot h$ .

What if you are given the sides of a triangle are 5 and 6 and the angle between the sides is $\frac{\pi}{3}$ ? 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 $h$  for height:

$\sin C & = \frac{h}{a}\\a \sin C & = h$

Substituting into the area formula:

$Area & = \frac{1}{2} b \cdot h\\Area & = \frac{1}{2} b \cdot a \cdot \sin C\\Area & = \frac{1}{2} \cdot a \cdot b \cdot \sin C$

Example A

Given  $\Delta ABC$ with $A = 22^\circ, b = 6, c =7$ . 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.

$Area & = \frac{1}{2} bc \sin A\\Area & = \frac{1}{2} \cdot 6 \cdot 7 \cdot \sin 22^\circ \approx 7.86 \ldots units^2$

Example B

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

Solution:

$Area & = \frac{1}{2} \cdot a \cdot b \cdot \sin C\\28 & = \frac{1}{2} \cdot 8 \cdot 9 \cdot \sin C\\\sin C & = \frac{28 \cdot 2}{8 \cdot 9}\\C & = \sin^{-1} \left( \frac{28 \cdot 2}{8 \cdot 9} \right) \approx 51.05 \ldots ^\circ$

Example C

Given triangle  $ABC$ with $A = 12^\circ, b = 4$  and  $Area = 1.7 \ un^2$ , what is the length of side $c$

Solution:

$Area & = \frac{1}{2} \cdot c \cdot b \cdot \sin A\\1.7 & = \frac{1}{2} \cdot c \cdot 4 \cdot \sin 12^\circ\\c & = \frac{1.7 \cdot 2}{4 \cdot \sin 12^\circ} \approx 4.08 \ldots$

Concept Problem Revisited

What if you are given the sides of a triangle are 5 and 6 and the angle between the sides is $\theta = \frac{\pi}{3}$

$Area = \frac{1}{2} \cdot 5 \cdot 6 \cdot \sin \frac{\pi}{3} \approx 12.99 \ un^2$

#### 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  $\Delta ABC$ with $A = 31^\circ, b = 12, c = 14$ ?

2. What is the area of  $\Delta XYZ$ with $x = 11, y = 12, z = 13$ ?

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.  $Area = \frac{1}{2} \cdot 12 \cdot 14 \cdot \sin 31^\circ \approx 43.26 \ldots units^2$

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 $52.02 \ldots^\circ$  therefore the area is:

$Area = \frac{1}{2} \cdot 12 \cdot 13 \cdot \sin 52.02 \ldots \approx 61.5 \ un^2$

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

$Area = \sqrt{s(s - a) (s - b) (s - c)}$

Where $s$  is the semi perimeter:

$s = \frac{a + b + c}{2}$

3.  $3 = \frac{1}{2} \cdot 4 \cdot 5 \cdot \sin \theta$

$\theta = \sin^{-1} \left( \frac{3 \cdot 2}{4 \cdot 5} \right) \approx 17.45 \ldots^\circ$

#### Practice

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

1. $\Delta ABC$  if $a=13, b=15$ , and $\angle C = 70^\circ$ .
2. $\Delta ABC$  if $b=8, c=4$ , and $\angle A = 58^\circ$ .
3. $\Delta ABC$  if $b=34, c=29$ , and $\angle A = 125^\circ$ .
4. $\Delta ABC$  if $a=3, b=7$ , and $\angle C = 81^\circ$ .
5. $\Delta ABC$  if $a=4.8, c=3.7$ , and $\angle B = 54^\circ$ .
6. $\Delta ABC$  if $a=12, b=5$ , and $\angle C = 22^\circ$ .
7. $\Delta ABC$  if $a=3, b=10$ , and $\angle C = 65^\circ$ .
8. $\Delta ABC$  if $a=5, b=9$ , and $\angle C = 11^\circ$ .
9. $\Delta ABC$  if $a=5, b=7$ , and $c=8$ .
10. $\Delta ABC$  if $a=7, b=8$ , and $c=14$ .
11. $\Delta ABC$  if $a=12, b=14$ , and $c=13$ .
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  $\Delta DEF$ has area 32 square inches, what is the angle included between side length 9 and 10?
15. Given  $\Delta GHI$ has area 15 square inches, what is the angle included between side length 7 and 11?

### Vocabulary Language: English

Included Angle

Included Angle

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

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.