# Trigonometry : Area of a Triangle

## Example Questions

### Example Question #1 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  cm, , and .

Explanation:

We don't have this triangle's height to use the formula , so we'll need to use trigonometry to find the area of this triangle. We can use the formula , where R is the area of a triangle. This formula comes from the fact that  and also incorporates triangle trigonometry and the Law of Sines. Applying the formula, we get:

### Example Question #2 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  cm, , and .

Explanation:

We don't have this triangle's height to use the formula , so we'll need to use trigonometry to find the area of this triangle. We can use the formula , where R is the area of a triangle. This formula comes from the fact that  and also incorporates triangle trigonometry and the Law of Sines. Applying the formula, we get:

### Example Question #2 : Find The Area Of A Triangle Using Trigonometry

Find the area of , given m, m, and .

or

or

or

or

or

or

Explanation:

Since we are not given the height of this triangle, we'll need to use trigonometry to find its area. If you draw and label the triangle, you'll notice that we are given Side-Side-Angle, which is the ambiguous case, and therefore there may be two different values that satisfy . Let's start by finding those.

Use the Law of Sines to find

Because this value is less than 1, there are two values that satisfy . We can now find these using inverse trig and knowing that . We have:

and

Because we have two values that satisfy , we'll also have two angles that satisfy , and two unique triangles with two unique areas.  Each of these is found below.

Next, we can use the formula , where R is the area of a triangle. This formula comes from the fact that  and also incorporates triangle trigonometry and the Law of Sines. Applying the formula, we get:

Now, find  for our second triangle.

### Example Question #4 : Find The Area Of A Triangle Using Trigonometry

Find the area of , given  feet, and  feet.

Explanation:

Since we are not given the height of this triangle, we'll need to use trigonometry to find its area. If you draw and label the triangle, you'll notice that we are given Side-Side-Angle, which is the ambiguous case, and therefore there may be two different values that satisfy . Let's start by finding those.

Use the Law of Sines to find

While this is the ambiguous case, there is only one solution if the side opposite the given angle (a=125 ft) is equal to or greater than the other given side (b=95 ft). Because , there is only one value of  and only one solution to this triangle's area. Therefore, we can continue onward to find this triangle's area.

Next, we can use the formula , where R is the area of a triangle. This formula comes from the fact that  and also incorporates triangle trigonometry and the Law of Sines. Since we know a and b both, we could use either of the first two formulas to find the area. You'll get the same answer either way; feel free to test the other one to see this shown. Applying the formula, we get:

### Example Question #3 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  inches,  inches, and .

Explanation:

Using the formula for area of a triangle equal to , drawing  and labelling its sides, angles, and height h, then using triangle trigonometry and substitution, we can derive the formulae , where R is equal to area. This can be used to find the area of a triangle when we know two of its sides and the included angle. Plugging in, we get:

Therefore the area of this triangle is 129.2 square inches.

### Example Question #6 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  cm,  cm, and .

Explanation:

Using the formula for area of a triangle equal to , drawing  and labelling its sides, angles, and height h, then using triangle trigonometry and substitution, we can derive the formulae , which can be used to find the area of a triangle when we know two of its sides and the included angle. Plugging in, we get:

Therefore, the area of the triangle is 95.38 square centimeters.

### Example Question #7 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  meters,  meters, and  meters.

15.998 meters

10.5 meters

234.359 meters

4.93 meters

15.998 meters

Explanation:

Because we do not know the height of this triangle, we need to lean into trigonometry. Luckily, Heron's Formula will help us solve this problem. You can use it when you know three sides of a triangle and want to find the area. Please note that in order to find a triangle's area, you must make sure that the triangle exists by satisfying the Triangle Inequality, which states that the sum of any two sides of the triangle must always be larger than the remaining side. This is true for the lengths 4, 8, and 9 given in this problem. Heron's formula can be described as:

where R is the area of the triangle and s is the semiperimeter equivalent to . Using these formulas and plugging in, we get:

Therefore the area of  is 15.998 meters.

### Example Question #8 : Find The Area Of A Triangle Using Trigonometry

Find the area of  given that  feet,  feet, and  feet.

3.36 feet

1.83 feet

43.32 feet

.86 feet