# Area of Pentagon

## Trigonometry # Area of Pentagon

The area of a pentagon is a region covered by all the sides of the pentagon. A pentagon is a two-dimensional shape in geometry. A pentagon is a five-sided polygon, also called 5-gon. It can be regular as well as irregular. The sides and angles of a regular pentagon are equal to each other.

The interior angle equals 108 degrees, and its exterior angle is equal to 72 degrees. The sum of interior angles of the pentagon is 540° and the sum of exterior angles is 360°.

In this article, we will learn the area of the pentagon for regular and irregular pentagon along with formulas and examples.

## What is Area of Pentagon?

The area of a pentagon is space covered by the sides of the pentagon in a plane. The unit of area of a pentagon is given by square unit, such as cm2, m2, in2, etc. ## Area of Pentagon Formula

Area of pentagon formula can be derived from three different methods.

• Using the length of the side (in case of the regular pentagon)
• Using the apothem
• Using the area of an isosceles triangle

### Area of Regular Pentagon

If the side of the regular pentagon is given, then the area of the pentagon is given by:

Area of a pentagon, $A = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^{2}$

where s is the side length of pentagon.

### Area of Pentagon using Apothem

If the side-length and apothem is given of a pentagon, then;

Area of Pentagon = 5/2 x s x a;

where ‘s’ is the side of the pentagon, and ‘a’ is the apothem length. Apothem is the line from the center of the pentagon to a side, intersecting the side at 90 degrees right angle.

### Area of Pentagon Using area of Isosceles triangles

If the pentagon is divided into five equal isosceles triangles, then we can find the area of an individual triangle and add them to get the area of the pentagon.

Area of pentagon = Sum of the area of five isosceles triangles formed within the pentagon

## How to Find Area of Pentagon?

To find the area of a pentagon, divide the regular pentagon into five equal triangles. Each of the triangles is an isosceles triangle. By the formula of area of the isosceles triangle, when all three sides are given,

Area = A = ½[√(a2 − b⁄4) × b]

where a is the length of equal sides and b is the base of the triangle.

Hence,

Area of Pentagon = 5 × Area of isosceles triangle

### Perimeter of Pentagon Formula

The perimeter of a pentagon is the total length of its boundaries. Therefore, it is equal to the sum of all its five sides. For a regular pentagon, the perimeter is given by: Example: If the side length of a regular pentagon is equal to 7 cm. Then find its perimeter.

Solution: Given, length of the side of the pentagon = 7 cm

Perimeter of pentagon = 5 x side = 5 x 7 = 35 cms

### Diagonals of Pentagon Formula

The number of diagonals in a Pentagon is five.

Number of diagonals in any pentagon = {n*(n-3)}/2, where n = number of sides

In the case of the pentagon, the number of sides is equal to 5, n = 5.

Therefore, Number of diagonals of pentagon = [5(5-3)]/2 = (5 x 2)/2 = 5

## Solved Examples on Area of Pentagon

Example 1: Let’s take the pentagon with a side length is 5 units and apothem length is 2 units

Area of pentagon is = 5/2 x s x a

= 5/2 x 5 x 2

=25 square centimetres.

Example 2: Find the area of a pentagon of side 5 cm and apothem length 3 cm?

Solution :

Given,

s = 5 cm

a = 3 cm

Area of a pentagon

= 37.5 cm

### Practice Questions on Area of Pentagon

1. What is the area of the pentagon if the side length of a regular pentagon is 7 cm?
2. If the apothem is 2.5 cm and the area of a pentagon is 100 sq.cm, find the side length of a pentagon.

## Frequently Asked Questions – FAQs

### What is the area of pentagon?

The area of a pentagon is the region covered by its five sides in a plane.

### What is the formula for area of pentagon with apothem?

If ‘s’ is the side and ‘a’ is the apothem, then the area of pentagon is equal to 5/2 x s x a.

### How to derive the area of pentagon?

To derive the area of pentagon, divide the regular pentagon into five equal isosceles triangles. Add the area of all the triangles to get the area of pentagon.

### Can we find the area of pentagon without knowing the length of its sides?

To find the area of pentagon, we have to know the length of its sides. Using apothem, we have area of pentagon equals to 5/2 x s x a. Using only sides of regular pentagon, the area will be 1/4[√5(5+2√5))]s2