Pentagon Calculator. Find the area, side a, diagonal and perimeter of a Pentagon with our free to use calculator. Fantastic math tool. Click below to utilise it right now.

## What is a pentagon?

We know that a pentagon is a polygon formed with five sides and five angles. Here, “Penta” denotes five and “gon” denotes angle. Also, in a regular pentagon, there are five symmetrical lines and it has rotational symmetry of order of 5.

## How do we find the area of a pentagon?

When we talk of the area of the pentagon we mean to consider the space that is confined within the sides of the polygon. The area of a pentagon is calculated by using the following formula –

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

where “ a “ is the side of a regular pentagon.

Let us understand the formula using an example.

**Example**

Suppose, we wish to find the area of a pentagon having a side of 4 cm

**Solution**

We have been given that we have a pentagon that has a side of 4 cm. Now, we already know that the area of a pentagon is given by Area of a pentagon = $\frac{1}{4}\sqrt{5(5+2\sqrt{5})}$ a^{2} , where “ a “ is the side of a regular pentagon.

If we substitute “ a ” in the above equation by 4, which is the length of a side of our regular pentagon, we will have,

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

⇒ Area of a pentagon = $\frac{1}{4}\sqrt{5(5+2\sqrt{5})}$ x 4 x 4

Now, the value of = 2.236

Substituting this value of in the above equation, we will have,

Area of a pentagon = $\frac{1}{4}\sqrt{5(5+2\sqrt{5})}$ x 4 x 4

⇒ Area of a pentagon = 27.5277 sq. cm

**Therefore, the area of the pentagon having a side of 4 cm will be 27.5277 sq. cm = 27.53 sq. cm**

How can we use the pentagon calculator to perform this operation? Let us find out.

## How to use the pentagon calculator to find the area of a pentagon when we are given its side?

We know that Area of a pentagon = $\frac{1}{4}\sqrt{5(5+2\sqrt{5})}$ a^{2}

where “ a “ is the side of a regular pentagon.

Through the above formula we can see that if are given the side of a regular pentagon, we can find its area.

Let us consider the example that we discussed above where we calculated the area of a pentagon having a side of 4 cm. We shall now see how this operation can be performed using the pentagon calculator. The following steps are to be followed for finding the area of a pentagon –

**Step 1** – The first step is to choose the option of solving for “ area “ from the drop-down option given in the “ Enter Information “ section. Below is a snapshot of how the drop-down box would like upon choosing the area as selection –

**Step 2** – Once we have chosen “ Solving for the area “ of our option, the next side is to choose what is given to us. In the given section, we need to choose “ side a “ as the option as we are given with the side of the pentagon. Below is a snapshot of how the drop-down box would like upon choosing side as selection –

**Step 3** – Now that we have chosen both what is required and what is given, we need to enter the value of the side that is given to us. As soon as we choose a side as the option in the previous step, another box crops up that asks us to enter the value of the side of the pentagon. Since we have been given the side as 4 cm, we shall enter 4 in this box. Below is a snapshot of how the drop-down box would like upon having the requirement of entering the value of the side –

**Step 4** – The last step is to click on the calculate button to obtain the value of the area. As soon as we will do that we shall have the results on the right hand side of the information that we entered in the above steps. Below is a snapshot of how the result would be displayed once we click on the calculate button.

We can clearly see that only do we get the answer we also get to know the formula used as well as the steps involved in the calculation. This not only helps us to verify our answers but also helps us in having getting to learn more about the concepts related to the area of the circle and other dimensions related to it.

Can we find the area of a pentagon using its perimeter? Let us find out.

## How to find the area of a pentagon when are given its perimeter?

In order to find the area of a pentagon when are given its perimeter, we need to first understand the formula of the perimeter of the pentagon.

**The perimeter of a pentagon = 5 x a, ****where, “ a “ is the side of a regular pentagon.**

Let us understand it using an example.

Suppose we have a pentagon having the side 4 cm. what would be its perimeter?

The perimeter of the pentagon would be given by 5 x Side = 5 x 4 = 20 cm.

Now, let us see how we can use this formula to find the area of a pentagon.

We know that

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

where “ a “ is the side of a regular pentagon.

We also know that the Perimeter ( P ) of a pentagon = 5 x a, where “ a “ is the side of a regular pentagon. This means that P = 5a ⇒ a = $\frac{P}{5}$ …………………….. ( 2 )

Substituting the value of s from ( 2 ) in equation ( 1 ) we get,

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

⇒ Area of a pentagon = $\frac{1}{4}\sqrt{5(5+2\sqrt{5})} x \frac{P x P}{25}$

⇒ **Area of a pentagon =** $\sqrt{5(5+2\sqrt{5})} x \frac{PxP}{100}$ which is our formula for finding the area of a pentagon using its perimeter.

Let us understand it using an example.

Suppose the perimeter of a pentagon is 25 cm and we want to find its area. We will have,

Area of a pentagon = $\sqrt{5(5+2\sqrt{5})} x \frac{PxP}{100}$

⇒ Area of a pentagon= $\sqrt{5(5+2\sqrt{5})} x \frac{25×25}{100}$ = 43.01 sq. cm (approx.)

Let us see how to use the pentagon calculator for finding the area using perimeter.

## How to use the pentagon calculator to find the area of a pentagon when we are given its perimeter?

The steps required to find the area of the pentagon when we have been given the perimeter are the same as we performed above except for the fact that in step 2 we will choose perimeter in the given section in step 2.

Below is the snapshot that displays the input as well as the result we will receive after entering the given input –