### All High School Math Resources

## Example Questions

### Example Question #10 : How To Find The Area Of An Equilateral Triangle

Determine the area of the following equilateral triangle:

**Possible Answers:**

**Correct answer:**

The formula for the area of an equilateral triangle is:

,

where is the length of the sides.

Plugging in our value, we get:

### Example Question #11 : How To Find The Area Of An Equilateral Triangle

Find the area of an equilateral triangle whose perimeter is

**Possible Answers:**

**Correct answer:**

The formula for the perimeter of an equilateral triangle is:

Plugging in our values, we get

The formula for the area of an equilateral triangle is:

Plugging in our values, we get

### Example Question #154 : Plane Geometry

What is the area of an equilateral triangle with a side length of 5?

**Possible Answers:**

**Correct answer:**

Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height.

If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5.

Therefore we can use the Pythagorean Theorem to solve for the height:

Now we can find the area of the triangle:

### Example Question #1 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle has a perimeter of 18. What is its area?

**Possible Answers:**

**Correct answer:**

Recall that an equilateral triangle also obeys the rules of isosceles triangles. That means that our triangle can be represented as having a height that bisects both the opposite side and the angle from which the height is "dropped." For our triangle, this can be represented as:

Now, although we do not yet know the height, we do know from our 30-60-90 regular triangle that the side opposite the 60° angle is √3 times the length of the side across from the 30° angle. Therefore, we know that the height is 3√3.

Now, the area of a triangle is (1/2)bh. If the height is 3√3 and the base is 6, then the area is (1/2) * 6 * 3√3 = 3 * 3√3 = 9√(3).

### Example Question #163 : Plane Geometry

A circle contains 6 copies of a triangle; each joined to the others at the center of the circle, as well as joined to another triangle on the circle’s circumference.

The circumference of the circle is

What is the area of one of the triangles?

**Possible Answers:**

**Correct answer:**

The radius of the circle is 2, from the equation circumference . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle

To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.

We know that the hypotenuse is 2 so . That's our solution. We know that the base is 1, and if you square 1, you get 1.

Now our formula looks like this: , so we're getting close to finding .

Let's subtract 1 from each side of that equation, in order to make things a bit simpler:

Now let's apply the square root to each side of the equation, in order to change into :

Therefore, the height of our equilateral triangle is

To find the area of our equilateral triangle, we simply have to multiply half the base by the height:

The area of our triangle is

### Example Question #1 : How To Find The Height Of An Equilateral Triangle

An equilateral triangle has a side length of . What is the triangle's height ?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

The altitude, , divides the equilateral triangle into two right triangles and divides the bottom side in half.

In a right triangle, the sides of the triangle equal , , and . In these equations equals the length of the smallest side, which in our triangle is or .

In this scenario:

and

Therefore,

### Example Question #2 : How To Find The Height Of An Equilateral Triangle

An equilateral triangle has a side length of . What is its height, ?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

An altitude slices an equilateral triangle into two triangles. These triangles follow a side-length pattern. The smallest of the two legs equals and the hypotenuse equals . By way of the Pythagorean Theorem, the longest leg or .

Therefore, we can find the height of the altitude of this triangle by designating a value to . The hypotenuse of one of the is also the side of the original equilateral triangle. Therefore, one can say that and .

### Example Question #3 : How To Find The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with side 6?

**Possible Answers:**

**Correct answer:**

When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the ) is .

### Example Question #4 : How To Find The Height Of An Equilateral Triangle

Find the height of the following equilateral triangle:

**Possible Answers:**

**Correct answer:**

Each angle in an equilateral triangle is .

Use the formula for triangles in order to find the length of the height.

The formula is:

Where is the length of the side opposite the

If we were to create a triangle by drawing the height, the length of the side is , the base is , and the height is .

### Example Question #5 : How To Find The Height Of An Equilateral Triangle

Solve for the value of X in the following equilateral triangle:

**Possible Answers:**

**Correct answer:**

If we draw a line segment between X and the base of the triangle, we form a triangle.

We can use the relationships between the sides of a triangle in order to find the length of X.

We know the base opposite the is .

The value of the height opposite the must then be , or .

Therefore, the value of X will be twice the value of the height:

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