The **perimeter of a pentagon** is the sum of all its sides. A pentagon is a polygon with five sides, five vertices, and five interior angles that add up to 540°. Pentagons can be regular or irregular depending upon the length of the sides. Let us study more about the perimeter of pentagon in this article.

1. | What is the Perimeter of a Pentagon? |

2. | Perimeter of Pentagon Formula |

3. | Perimeter of Pentagon with Radius |

4. | FAQs on Perimeter of Pentagon |

## What is the Perimeter of a Pentagon?

The perimeter of a pentagon is the total measure of its boundary and it is calculated by adding the length of all its sides. Since a pentagon is a five-sided polygon, all the five sides can be added to find its perimeter. The perimeter of a pentagon is expressed in linear units like inches, yards, centimeters, and so on.

### Regular and Irregular Pentagon

A regular pentagon is a pentagon in which all five sides are of equal length and all the interior angles are of equal measure. An irregular pentagon is one in which the sides are not equal and the angles are not of equal measure.

## Properties of a Regular Pentagon

The word 'pentagon' is derived from the Greek word 'Penta' which means 'five', and 'gon' which means 'angle'. There are some basic properties that help us to identify a pentagon. Observe the following regular pentagon to relate to its properties given below.

We can identify and distinguish a regular pentagon with the help of the following properties:

- Pentagon is a polygon with 5 sides.
- It has 5 diagonals.
- The sum of all the interior angles of a pentagon is 540°
- Each interior angle of a regular pentagon is 108°
- Each exterior angle of a regular pentagon is 72°

Now, let us read about the formulas that are used to find the perimeter of a pentagon. The formulas differ according to the known dimensions.

## Perimeter of Pentagon Formula

We know that the perimeter of a pentagon is the sum of the lengths of all its sides. Now, if it is a regular pentagon, it means all five sides are of equal length. Let us consider a regular pentagon of side length 'a'. In this case, the perimeter can be calculated with the help of the formula, Perimeter = a + a + a + a + a, which is 5a. Therefore, the basic formula for the perimeter of a regular pentagon is given below.

**Perimeter of pentagon = Sum of all its sides**

**Example 1:** Find the perimeter of a regular pentagon that has a side length of 7 units.

**Solution:**

The side length (a) of the given pentagon = 7 units.

Using the perimeter of pentagon formula, **Perimeter = 5a** = 5 × 7 = 35 units.

**Example 2:** Find the perimeter of a pentagon in which the length of the sides are 5 units, 4 units, 7 units, 3 units, and 6 units.

**Solution:**

In order to find the perimeter of the irregular pentagon, we will add all the five sides, perimeter = 5 + 4 + 7 + 3 + 6 = 25 units.

Therefore, the perimeter of the pentagon is 25 units.

## Perimeter of Pentagon with Radius

The perimeter of a pentagon can be calculated if the radius of the pentagon is given. This is the case when a pentagon is inscribed in a circle as shown below. In such cases, the radius is the line drawn from the center of the polygon to one of its vertices. It is also referred to as the circumradius. The perimeter of this pentagon can be calculated once the side length is known with the help of the formula: Side length = 2r × Sin(180/n), where 'r' is the radius and 'n' represents the number of sides. After getting the side length, the perimeter can be easily calculated with the formula, Perimeter = 5a.

**Example:** Find the perimeter of a regular pentagon inscribed in a circle with a radius of 5 units.

**Solution: **Given, radius = 5 units, number of sides = 5

Let us first calculate the side length of the pentagon with the help of the formula: Side length = 2r × Sin(180/n)

On substituting the given values in the formula, we get, Side length = 2r × Sin(180/n) = Side length = 2 × 5 × Sin(180/5) = 10 × Sin(180/5) = 10 × Sin 36 = 10 × 0.58779 = 5.8 units. After this, the perimeter of the regular pentagon can be calculated using the formula, Perimeter = 5a = 5 × 5.8 = 29 units.

**☛ Related Articles**

- Perimeter Formulas
- Perimeter of Polygon
- Difference Between Area and Perimeter
- Perimeter of Rectangle
- Perimeter of Square
- Perimeter of Triangle

## FAQs on Perimeter of Pentagon

### What is the Perimeter of a Pentagon?

The perimeter of a pentagon is the total measure of its boundary and it is calculated by adding the length of all its sides. If it is a regular pentagon the formula to find its perimeter is, P = 5a, where 'a' is the length of the side. In the case of an irregular pentagon, where all the sides are of different lengths, the formula is, P = sum of all its sides. The perimeter of a pentagon is expressed in linear units like inches, yards, centimeters, and so on.

### How to Find the Perimeter of a Pentagon?

The perimeter of a pentagon can be calculated with the help of the basic formula, Perimeter of pentagon = sum of its sides. Sometimes, the side length of a regular pentagon is not known and other dimensions like the radius are given. In such cases, the side length is calculated with the help of the formula, Side length = 2r × Sin(180/n), where 'r' is the radius and 'n' represents the number of sides. After getting the side length, the perimeter can be calculated with the formula, Perimeter = 5a.

### What is the Formula of the Perimeter of a Pentagon?

The basic formula that is used to find the perimeter of pentagon is, P = sum of its sides. If it is a regular pentagon, the formula can be written as, P = 5a, where 'a' is the length of its side. If it is an irregular pentagon, all its sides can be added to get the perimeter.

### How to Find the Perimeter of a Pentagon with Radius?

The perimeter of pentagon can be calculated if the radius of the pentagon is given. This is the case when a pentagon is inscribed in a circle. In such cases, the radius is the line drawn from the center of the polygon to one of its vertices. It is also referred to as the circumradius. The perimeter of this pentagon can be calculated after finding the side length with the help of the formula: Side length = 2r × Sin(180/n), where 'r' is the radius and 'n' represents the number of sides. After getting the side length, the perimeter can be calculated with the formula, Perimeter = 5a.

### How to Find the Perimeter of a Pentagon with Apothem?

The perimeter of a pentagon can be calculated if the apothem of the pentagon is given. Apothem is a perpendicular line segment drawn from the center of the pentagon to the side of the pentagon. If the apothem of a pentagon is given, the side length can be calculated with the formula, Side length = 2 × apothem length × tan(180/n), where 'n' represents the number of sides. After getting the side length, the perimeter of the pentagon can be calculated with the formula, P = 5a.

### How to Find the Perimeter of a Pentagon with Area?

The perimeter of a regular pentagon can be calculated if the area and the apothem of a pentagon is given. The area and the apothem can help to find the side length of the pentagon with the help of the formula, Area = 5/2 × side length × apothem. After substituting the values in the formula the side length can be calculated. Once the side length is known, the perimeter of the pentagon can be found with the formula, Perimeter = 5 × side length

### How to Find the Perimeter of a Pentagon with Different Sides?

If the sides of a pentagon are not equal in measure, it is termed as an irregular pentagon. In this case, the perimeter of the pentagon can be calculated by using the basic formula, Perimeter = sum of its sides. For example, the different sides of a pentagon are of the lengths, 3 units, 4 units, 2 units, 5 units and 6 units, the perimeter of the pentagon will be, P = 3 + 4 + 2 + 5 + 6 = 20 units.

### How to Find the Perimeter of a Regular Pentagon?

If all the sides of a pentagon are of equal length, it is known as a regular pentagon. In this case, the perimeter of the pentagon can be calculated with the help of the formula, Perimeter = 5 × side length. For example, if one side of a regular pentagon is 8 units, its perimeter will be, P = 5 × 8 = 40 units.