# Intermediate Geometry : How to find the area of an equilateral triangle

## Example Questions

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

ΔABC is an equilateral triangle with side 6.

Find the area of ΔABC (to the nearest tenth).

18.7

13.1

14.2

7.2

15.6

15.6

Explanation:

Equilateral triangles have sides of all equal length and angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle =

Given the height, we can now find the area of the triangle using the equation:

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

ΔABC is an equilateral triangle with side 17.

Find the area of ΔABC (to the nearest tenth).

128.3

131.4

129.8

125.1

132.9

125.1

Explanation:

Equilateral triangles have sides of equal length, with angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle =

Given the height, we can now find the area of the triangle using the equation:

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

If the perimeter of an equilateral triangle is 54 inches, what is the area of the triangle in square inches?

Explanation:

To find the area you would first need to find what the length of each side is: 54 divided by 3 is 18 for each side.

Then you would need to draw in the altitude of the triangle in order to get its height.  Drawing this altitude will create two 30-60-90 degree triangles as shown in the picture.  The longer leg is  times the short leg.  Thus the height is

Next we plug in the base and the height into the formula to get

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

What is the area of this triangle if ?

Explanation:

The formula for the area of an equilateral triangle with side length  is

So, since ,

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

If the sides of this triangle are doubled in length, what is the triangle's new area in terms of the original length of each of its sides,  ?

Explanation:

The formula of the area of an equilateral triangle is  if  is a side.

Since the sides of our triangle have doubled, they have changed from  to . We can substitute  into the equation and solve for the triangle's new area in terms of :

### Example Question #6 : How To Find The Area Of An Equilateral Triangle

Suppose we triple the sides of this equilateral triangle to . What is the area of the new triangle in terms of ?

Explanation:

The formula for the area of an equilateral triangle is  if  is the length of one of the triangle's sides.

In this problem, the length of one of the triangle's sides is being tripled, so we can substitute  into the equation for  and solve for the triangle's new area in terms of :

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

What is half the area of the above triangle if ?

Explanation:

The formula for the area of an equilateral triangle is. For this problem's triangle, , so we can substitute  into the equation for  and solve for the area of the triangle:

At this point, we need to divide by , since the problem asks for half of the triangle's area:

### Example Question #8 : How To Find The Area Of An Equilateral Triangle

If , what is the area of the triangle?

Explanation:

We know this triangle is equilateral since each of its sides has the same length, . The formula of the area of an equilateral triangle is  if  is the length of one of the triangle's sides.

Since our side length is , we can substitute that value into the equation for  and solve for the area of the triangle:

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

If  , what is the area of this equilateral triangle?

Explanation:

Given that the sides of our equilateral triangle are each  long, we can just plug the value in to the formula for the area of an equilateral triangle and solve for the area of the triangle:

if  is a side of the triangle.

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

An equilateral triangle has sides of length 6cm.  If the height of the triangle is 4.5cm what is the area of the triangle?