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

## Example Questions

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

Find the area of an equilateral triangle with a perimeter of 24cm. Leave answer in simplest radical form.

Explanation:

To find the area of an equilateral triangle, one must find the base and the height.

All the sides of an equilateral triangle are congruent, so if the perimeter of the equilaterail triangle is 24, then each side must equal one third of that total which is 8cm.

This will produce a triangle that includes the following information below:

Dropping an altitude down the center of the equilateral triangle will result in two 30-60-90 triangles with a hypotenuse of 8.

In every 30-60-90 triangle the following formulas apply:

When we plug in the given information on the triangle we get:

Dividing both sides by 2 gives the below result.

We can now plug this into the long leg formula to get the height of the triangle:

Now that we have all of the information needed to find the area we plug these values into the area formula.

### Example Question #32 : Equilateral Triangles

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown in the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

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

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown in the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

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

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown in the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

### Example Question #35 : Equilateral Triangles

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown in the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

### Example Question #36 : Equilateral Triangles

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown in the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

### Example Question #681 : Intermediate Geometry

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown by the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

### Example Question #38 : Equilateral Triangles

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown by the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

### Example Question #241 : Triangles

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown by the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.

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

A circle with a radius of  is inscribed in an equilateral triangle with side lengths of  as shown by the figure below.

Find the area of the shaded region.

Explanation:

In order to find the area of the shaded region, we must first find the area of the circle and the area of the equilateral triangle.

Recall how to find the area of a circle:

Plug in the given radius to find the area of the circle.

Next, recall how to find the area of an equilateral triangle:

Plug in the length of the side of the triangle to find the area.

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

Make sure to round to  places after the decimal.