# Intermediate Geometry : How to find the area of a pentagon

## Example Questions

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### Example Question #21 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Explanation:

Recall that the area of regular polygons can be found using the following formula:

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

Substitute in the length of the given side of the pentagon in order to find the perimeter.

Next, substitute in the given and calculated information to find the area of the pentagon.

### Example Question #22 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Explanation:

Recall that the area of regular polygons can be found using the following formula:

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

Substitute in the length of the given side of the pentagon in order to find the perimeter.

Next, substitute in the given and calculated information to find the area of the pentagon.

### Example Question #23 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Explanation:

Recall that the area of regular polygons can be found using the following formula:

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

Substitute in the length of the given side of the pentagon in order to find the perimeter.

Next, substitute in the given and calculated information to find the area of the pentagon.

### Example Question #24 : How To Find The Area Of A Pentagon

Find the area of the regular pentagon.

Explanation:

Recall that the area of regular polygons can be found using the following formula:

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

Substitute in the length of the given side of the pentagon in order to find the perimeter.

Next, substitute in the given and calculated information to find the area of the pentagon.

### Example Question #25 : How To Find The Area Of A Pentagon

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

Explanation:

The formula for area of a pentagon is , with representing the length of one side and representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

Each of these triangles have angle measures of , with being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and .

To solve for the apothem, we  can use basic trigonometric ratios:

Now that we know the apothem length, we can plug in all our values to solve for area:

### Example Question #26 : How To Find The Area Of A Pentagon

Each side of this regular pentagon has a length of . Solve for the area of the pentagon. Round to the nearest tenth.

Explanation:

When given the value of one side of a regular pentagon, we can assume all sides to be of equal length and we can use this formula to calculate area:

For thi formula, represents the length of one side while represents the number of sides. Therefore, we would plug in the values as such:

### Example Question #27 : How To Find The Area Of A Pentagon

A regular pentagon has a side length of . Find the area rounded to the nearest tenth.

Explanation:

We can use the following equation to solve for the area of a regular polygon with representing side length and representing number of sides:

### Example Question #28 : How To Find The Area Of A Pentagon

In the figure below, find the area of the pentagon if the length of  is one-fourth the length of .

Explanation:

Start by figuring out the lengths of  and .

Let  be the length of . From the question, we then write the length of  as .

From the figure, you should see that .

Plug in the variables and solve for .

So then the length of  must be , and the length of  must be .

Now, to find the area of the pentagon, notice that we can break the shape down to one rectangle and two right triangles.

Next, find the area of rectangle .

Next, find the area of the right triangles.

For triangle ,

For triangle ,

Add up the component areas to find the area of the pentagon.

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