# Trigonometry : 30-60-90 Triangles

## Example Questions

### Example Question #1 : 30 60 90 Triangles

In a 30-60-90 triangle, the length of the side opposite the  angle is . What is the length of the hypotenuse?

Explanation:

By definition, the length of the hypotenuse is twice the length of the side opposite the angle.

Recall that the hypotenuse is the side opposite the  angle.

Thus, using the equation below, where ss represents the short side (that opposite the angle) we get:

Plugging in our values for the short side we find the hypotenuse as follows:

### Example Question #2 : 30 60 90 Triangles

A triangle has three angles ,  and  such that  and . The side opposite to  measures  units in length. How long is the side opposite of ?

Explanation:

A triangle with a  angle relation is a , ,  degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle,  units.

Therefore, to make the above statement true .

### Example Question #3 : 30 60 90 Triangles

Triangle  is equilateral with a side length of .

What is the height of the triangle?

Explanation:

An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,

so..

### Example Question #4 : 30 60 90 Triangles

In a  triangle, the side opposite the  degree angle is . How long is the side opposite the  degree angle?

Explanation:

Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.

Therefore,

### Example Question #5 : 30 60 90 Triangles

What is the height of an equilateral triangle with side length 8?

Explanation:

The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is .

To double check the answer use the Pythagorean Thereom:

### Example Question #6 : 30 60 90 Triangles

What is the ratio of the side opposite the  angle to the hypotenuse?

Explanation:

Step 1: Locate the side that is opposite the  side..

The shortest side is opposite the  angle. Let's say that this side has length .

Step 2: Recall the ratio of the sides of a  triangle:

From the shortest side, the ratio is .

is the hypotenuse, which is twice as big as the shortest side..

The ratio of the short side to the hypotenuse is

### Example Question #7 : 30 60 90 Triangles

It is known that the smallest side of a 30-60-90 triangle is 5.

Find .

Explanation:

We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.

Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.

The formula for

, so  or .

### Example Question #8 : 30 60 90 Triangles

It is known that for a 30-60-90 triangle,

.

Find the area of the triangle.

Note:

Explanation:

First, we know that in a 30-60-90 triangle,

.

Also, the base is the smallest side times , so in our case it is .

The height is just the smallest side, .

Substituting these values into the formula given for area of a triangle, we obtain the answer .