# Area of a Regular Polygon Calculator

Created by Hanna Pamuła, PhD candidate
Reviewed by Bogna Szyk
Last updated: May 09, 2022

This area of a regular polygon calculator can help - as you can guess - in determining the area of a regular polygon. Type the number of sides and the polygon area appears in no time. If you're wondering how to find the area of a polygon formula, keep reading and you'll find the answer! If you want to calculate the area of any 3-sided or 4-sided polygon, check out this triangle area and quadrilateral area calculator.

## Area of a regular polygon formulas

The most popular, and usually the most useful formula is the one that uses the number of sides $n$ and the side length $a$:

• $\text{area} = n \times a^2 \times \cot{(π/n)} / 4$

However, given other parameters, you can also find out the area:

• $area = n \times a \times ri / 2$, having $ri$ - incircle radius (it's also an apothem - a line segment from the center to the midpoint of one of its sides)
• $\text{area} = \text{perimeter}\times ri / 2$, given $ri$ and polygon perimeter
• $\text{are}a = n \times ri^2 \times \tan{(π/n)}$, given $ri$
• $\text{area} = n \times rc^2 \times \sin{(2π/n)} / 2$, having $rc$ - circumcircle radius

## How to find the area of a polygon?

If you want to find the area of a regular polygon, simply use the formulas described above. However, if the polygon is not regular (what means it isn't equiangular and equilateral), you can:

1. Find the area using vertices coordinates:

$\small\text{area} = \left|\sum (x_i \times y_{i+1}) - (y_i \times x_{i+1})\right| / 2$

with $\small x(n+1) \rightarrow x(1)$ and $\small y(n+1) \rightarrow y(1)$

1. Determine the polygon area given side lengths and some diagonals, by splitting the polygon into triangles. Then find the area with given three sides (SSS) equation (it's called Heron's formula).

2. Calculate the area of polygons using other formulas - e.g. for a scalene triangle or a quadrilateral.

## Using the area of regular polygon calculator: an example

Let's assume that you want to calculate the area of a specific regular polygon, e.g. 12-sided polygon, dodecagon with 5-inch sides.

1. Enter the number of sides of chosen polygon. Put 12 into the number of sides box.
2. Type in the polygon side length. In our example, it's equal to 5 in.
3. Our area of polygon calculator displays the area. It's 279.9 in².

If you want to calculate the area of a regular polygon using other parameters than the side length, check out this general regular polygon calculator.

## FAQ

### How do I calculate the area of a regular polygon given the radius?

To calculate the area of a regular polygon given the radius apply the formula:
area = n * a² * cot(π/n) / 4
Where:

• n is the number of sides of the polygon;
• a is the length of the side; and
• cot is the cotangent function (cot(x) = 1/tan(x)).

### What is the area of a pentagon with side 3?

15.484. To compute the area of a pentagon with side 3, you can directly apply the formula:
area = n × a² × cot(π/n) / 4 ,
substituting:

• n = 5; and
• a = 3.

### What is the formula for the area of a polygon given the apotheme?

If you know the apothem of a regular polygon you can compute the area with the formula:
area = ap² × n × tan(π/n)
Where:

• ap is the apothem (the segment connecting the side midpoint to the center); and
• n is the number of sides of the regular polygon.

### What is the area of a hexagon with side 0.7621 m?

1.509 m². To calculate the area, use the formula for the area of a regular hexagon:
area = 6 × a² × cot(π/6) / 4
Where a is the side length.
This particular hexagon is equal in size to any one of the 18 elements in the primary mirror of the James Webb Space Telescope.

Hanna Pamuła, PhD candidate
Number of sides
Side a
in
in
in
Area
A
in²
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