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Pentagon Area Calculator

Introduction

Welcome to our Pentagon Area Calculator! which will help you calculate the Pentagon’s area quickly and accurately. A Pentagon is a two-dimensional closed figure with five sides and also five angles. The five interior angles in a pentagon add up to 540 degrees.

The space occupied in the two-dimensional plane by the pentagon is called the area of the pentagon.

Generally, we measure the pentagon’s area in square units, and we use units including sq cm, sq m, sq feet, sq inch, etc.

How to use the Pentagon Area Calculator?

Using the Pentagon Area calculator, you can calculate the area of a regular pentagon using two methods.

Area of Pentagon using Side and Apothem

You can calculate the area of the pentagon using the length of the side and apothem using the following formula

The variables in the Pentagon Area Calculator include:

Side (s)
Length of the side of the pentagon

Apothem (h)
The length of the perpendicular line from the center to one of the pentagon’s sides.

Area of the Regular Pentagon (A)
You can calculate the area of the pentagon using the following formula

A=12×Perimeter×Apothem=12×5s×h\begin{aligned} \text{A} &= \normalsize \dfrac{1}{2} \times \text{Perimeter} \times \text{Apothem} \\[10pt] &= \normalsize \dfrac{1}{2} \times 5s \times h \end{aligned}

Where,

s = length of the side of a regular pentagon

h = length of the perpendicular line from the center to one of the pentagon’s sides.

Area of Regular Pentagon using Side

We can also calculate the area of the Pentagon by using the length of the side using the following formula

The variables in the calculator include:

Side (s)
Length of the side of the pentagon

Area of the Regular Pentagon (A)
You can calculate the area of the pentagon using the following formula

A=14×5(5+25)×s2\text{A} = \normalsize \dfrac{1}{4} \times \sqrt{5(5+2\sqrt{5})} \times s^2

Where,

s = length of the side of a regular pentagon

What is a Pentagon?

The pentagon is a polygon with five sides and five angles. It is a closed two-dimensional shape. Basically, Pentagons can be classified into four types based on the length of their sides, a measure of their angles, and their vertices.

  1. Regular Pentagon: if all five sides of the Pentagon are equal and all five angles are equal, then the Pentagon is called a regular pentagon.
  2. Irregular Pentagon: if the sides and the angles of the Pentagon are not equal, then it is called an Irregular Pentagon.
  3. Convex Pentagon: if all of the Pentagon’s vertices point outward, then it is called a Convex Pentagon.
  4. Concave Pentagon: if at least one of the Pentagon’s vertex points inward, it is called a Concave Pentagon.

Properties of a Pentagon

  1. The sides of the Pentagon do not overlap.
  2. The sum of a Pentagon’s interior angles will equal 540 degrees.
  3. Each interior angle of a Regular Pentagon measures 108 degrees.
  4. Each exterior angle of a Regular Pentagon measures 72 degrees.

How is the area of the Pentagon Calculated?

The total space taken up by the Pentagon in a two-dimensional plane is called the area of the Pentagon. It could also be thought of as the number of square units required to fill the region inside the Pentagon. Hence, the measurement will be in square units.

The area of the Pentagon can be calculated in two ways:

Area of Pentagon using Side and Apothem

You can calculate the area of the pentagon using the length of the side and apothem using the following formula

A=12×Perimeter×Apothem=12×5s×h\begin{aligned} \text{A} &= \normalsize \dfrac{1}{2} \times \text{Perimeter} \times \text{Apothem} \\[10pt] &= \normalsize \dfrac{1}{2} \times 5s \times h \end{aligned}

Where,

s = length of the side of a regular pentagon

h = Apothem, the length of the perpendicular line from the center to one of the pentagon’s sides. This is also called the incircle radius.

Area of Regular Pentagon using Side

We can also calculate the area of the Pentagon by using the length of the side using the following formula

A=14×5(5+25)×s2\text{A} = \normalsize \dfrac{1}{4} \times \sqrt{5(5+2\sqrt{5})} \times s^2

Where,

s = length of the side of a regular pentagon

Examples

Example 1

Let’s say there is a Pentagon with a length of side equal to 16 cm, and the apothem, i.e. the perpendicular height from the center of the pentagon to one of the sides, is 11.01 cm. Find the area of the Pentagon.

The area of the Pentagon can be calculated using the following formula:

A=12×Perimeter×Apothem=12×5s×h=12×5×16×11.01=440.4  cm2\begin{aligned} \text{A} &= \normalsize \dfrac{1}{2} \times \text{Perimeter} \times \text{Apothem} \\[10pt] &= \normalsize \dfrac{1}{2} \times 5s \times h \\[10pt] &= \normalsize \dfrac{1}{2} \times 5 \times 16 \times 11.01 \\[10pt] &= 440.4 \;cm^2 \end{aligned}

Therefore the area of the Pentagon is 440.4 square cm.

Example 2

Let’s say there’s a Pentagon with a length of the side as 10 centimeters. What is the area of the Pentagon?

We can calculate the area of the Pentagon using the following formula.

A=14×5(5+25)×s2=14×5(5+25)×102≈172.05  cm2\begin{aligned} \text{A} &= \normalsize \dfrac{1}{4} \times \sqrt{5(5+2\sqrt{5})} \times s^2 \\[10pt] &= \normalsize \dfrac{1}{4} \times \sqrt{5(5+2\sqrt{5})} \times 10^2 \\[10pt] &\approx 172.05 \; cm^2 \end{aligned}

Therefore, the area of the Pentagon is approx. 172 square cm.

Author

hexacalculator design team

Our team blends expertise in mathematics, finance, engineering, physics, and statistics to create advanced, user-friendly calculators. We ensure accuracy, robustness, and simplicity, catering to professionals, students, and enthusiasts. Our diverse skills make complex calculations accessible and reliable for all users.

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