# Area of a Regular Polygon Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Nov 22, 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, along with a known property, and the polygon area will appear in no time. If you're wondering how to find the area of a polygon or what is the area of a polygon formula, keep reading. If you want to calculate the area of any 3-sided or 4-sided polygon, check out the triangle area calculator and the 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 $a$:

$A = n \times a^2 \times \frac{1}{4}\cot\left(\frac{\pi}{n}\right)$

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

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

🙋 The circumcircle is the circle that passes through all the polygon's vertex: you can learn how to calculate its center in the case of a triangle at our circumcenter of a triangle calculator.

## 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 (which means it isn't equiangular and equilateral), you can:

1. Find the area using vertices coordinates:
$A\! =\! \frac{\left|\sum (x_i \!\times\! y_{i+1})\! -\! (y_i \!\times\! x_{i+1})\right|}{2}$

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

1. Determine the polygon area given side lengths and some diagonals by splitting the polygon into triangles. Then find the area with the given three sides (SSS) equation (you can learn the origin of this formula with our Heron's formula calculator).

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., a 12-sided polygon, a dodecagon with 5-inch sides.

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

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