Area of Regular Polygon Calculator
Welcome to our Area of a Regular Polygon Calculator page.
We explain how to find the area of a regular polygon and provide a quick calculator to work it out for you, step-by-step.
You will also find formulas for finding the area of any regular polygon with up to 12 sides, and a handy printable sheet will all the information you need.
There are also some worked examples and worksheets for you to practice this skill.
Quicklinks to ...
Area of Regular Polygon Calculator
Our area of regular polygon calculator finds the area of a regular polygon when the number of sides and side lengths are known.
How the Calculator Works
- Choose the number of sides of the regular polygon you need.
- Choose the length and width of the regular polygon: you can choose a whole number, decimal or fraction.
- You can type a fraction by typing the numerator then '/' then the denominator.
- You can type a mixed number by typing the whole-number part, then a space then the fraction part.
- Examples: 2 1/2 (two and one-half); 3 4/5 (three and four-fifths); 7 1/3 (seven and one-third).
- Choose your units of measurement (default is none)
- Choose your desired accuracy (default is 2 decimal places)
- Click the Find Area button
- You will be shown the area as a decimal (and also a fraction if you typed the length as a fraction).
What is a Regular Polygon?
A polygon is the name of any 2 dimensional closed shape with straight sides. Polygons have at least 3 sides.
A regular polygon is a special type of polygon where the angles and the side lengths are all equal.
The shape below is a regular pentagon.
- It is a pentagon because it has 5 straight sides.
- It is regular because all the sides and angles are equal.
For more information on regular shapes, use this link.
Area of Regular Polygon Formula
The general formula for the area of a regular polygon is: \[ A = {ns^2 \over 4 \tan ({180^{\circ} \over n})} \]
This can also be written: \[ A = {n \over 4 } s^2 \cot ({180^{\circ} \over n}) \]
where n is the number of sides of the polygon, and s is the length of each side.
Area of Individual Regular Polygons
It is usually much better to use the individual formulas for different polygon, as they are usually simpler than using the formula above.
Our area of regular polygon calculator uses the individual formula below to calculate the area of each regular polygon.
| Regular Polygon | Shape Image | Area Formula |
|---|---|---|
|
Equilateral Triangle 3 sides |
|
\[ A = {\sqrt {3} \over 4} s^2 \] |
|
Square 4 sides |
|
\[ A = s^2 \] |
|
Regular Pentagon 5 sides |
|
\[ A = {1 \over 4} s^2 \sqrt {5(5 + 2 \sqrt 5)} \] |
|
Regular Hexagon 6 sides |
|
\[ A = {3 \sqrt {3} \over 2} s^2 \] |
|
Regular Heptagon 7 sides |
|
\[ A = {7 \over 4} s^2 \cot ({180^{\circ} \over 7}) \] |
|
Regular Octagon 8 sides |
|
\[ A = 2 s^2 (1 + \sqrt 2) \] |
|
Regular Nonagon 9 sides |
|
\[ A = {9 \over 4} s^2 \cot {20^{\circ}} \] |
|
Regular Decagon 10 sides |
|
\[ A = {5 \over 2} s^2 \sqrt {5 + 2 \sqrt 5} \] |
|
Regular Hendecagon 11 sides |
|
\[ A = {11 \over 4} s^2 \cot ({180^{\circ} \over 11}) \] |
|
Regular Dodecagon 12 sides |
|
\[ A = 3 s^2 (2 + \sqrt 3) \] |
Area of Regular Polygon Formula Sheet
We have taken the information from the table above and put it all on one easy to use simple sheet.
This sheet contains all the formula for finding the area of different regular polygons, from equilateral triangles to dodecagons.
The sheet also shows the general formala for finding the area of any regular polygon.
Area of Regular Polygon Examples
Area of a Regular Polygon Example 1
Find the area of the equilateral triangle below. Give your answer to 2 decimal places.
The equilateral triangle has a side length of 8 cm.
The area of an equilateral triangle is:
\[ A = { \sqrt 3 \over 4} s^2 \] where s is the length of each side.
So if we substitute the value of the side length into this equation, we get: \[ A = { \sqrt 3 \over 4} \cdot 8^2 = { \sqrt 3 \over 4} \cdot 64 = 16 \sqrt {3} \]
The area of the equilateral triangle is 27.71 cm2 to 2 decimal places.
Area of a Regular Polygon Example 2
Find the area of the regular hexagon below. Give your answer to 2 decimal places.
The regular hexagon has a side length of 6 m.
The area of a regular hexagon is:
\[ A = {3 \sqrt {3} \over 2} s^2 \] where s is the length of each side.
So if we substitute the value of the side length into this equation, we get: \[ A = {3 \sqrt {3} \over 2} \cdot 6^2 = {3 \sqrt {3} \over 2} \cdot 36 = 54 \sqrt 3 \]
The area of the regular hexagon is 93.53 m2 to 2 decimal places.
Area of a Regular Polygon Example 3
Find the area of the regular decagon below. Give your answer to 2 decimal places.
The regular decagon has a side length of 5 cm.
The area of a regular decagon is:
\[ A = {5 \over 2} s^2 \sqrt {5 + 2 \sqrt 5} \] where s is the length of each side.
So if we substitute the value of the side length into this equation, we get: \[ A = {5 \over 2} \cdot 5^2 \cdot \sqrt {5 + 2 \sqrt 5} = {5 \over 2} \cdot 25 \cdot \sqrt {5 + 2 \sqrt 5} \]
Simplifying this gives us: \[ A = {125 \over 2} \cdot \sqrt {5 + 2 \sqrt 5} = {125 \over 2} \cdot 3.077683 \]
The area of the regular decagon is 192.36 cm2 to 2 decimal places.
Area of a Regular Polygon Example 4
Find the area of the regular decagon below. Give your answer to 1 decimal place.
The regular heptagon has a side length of 9 ft.
The area of a regular heptagon is:
\[ A = {7 \over 4} s^2 \cot ({180^{\circ} \over 7}) \] where s is the length of each side.
So if we substitute the value of the side length into this equation, we get: \[ A = {7 \over 4} \cdot 9^2 \cdot \cot ({180^{\circ} \over 7}) = {7 \over 4} \cdot 81 \cdot \cot (25.714^{\circ}) \]
Simplifying this gives us: \[ A = {567 \over 4} \cdot \cot (25.714^{\circ}) = {567 \over 4} \cdot 2.076521 \]
The area of the regular heptagon is 294.3 ft2 to 1 decimal place.
Area of a Regular Polygon Example 5
Find the area of the regular pentagon below. Give your answer to 1 decimal place.
The regular pentagon has a side length of 4 ½ inches.
The area of a regular pentagon is:
\[ A = {1 \over 4} s^2 \sqrt {5(5 + 2 \sqrt 5)} \] where s is the length of each side.
So if we substitute the value of the side length into this equation, we get: \[ A = {1 \over 4} \cdot (4 {1 \over 2})^2 \cdot \sqrt {5(5 + 2 \sqrt 5)} = {1 \over 4} \cdot {81 \over 4} \cdot \sqrt {5(5 + 2 \sqrt 5)} \]
Simplifying this gives us: \[ A = {81 \over 16} \cdot \sqrt {5(5 + 2 \\sqrt 5)} = {81 \over 16} \cdot 6.8819096 \]
The area of the regular pentagon is 34.8 in2 to 1 decimal place.
Area of a Regular Polygon Worksheets
We have created two worksheets for you to practice the skill of finding the area of a range of different regular polygons.
Both sheets involve using the formulas above to find the area of different regular polygons where the side length is given.
You can use our area of regular polygon calculator to check you working out if you get stuck!
More Recommended Math Worksheets
Take a look at some more of our worksheets similar to these.
Regular Shapes Information
If you want to know more about regular shapes, including helpful information and printable shape sheets, take a look below.
More Area Support and Resources
We have a range of other area worksheets and support pages for a range of different 2d shapes.
More Area & Volume Calculators
We have a range of area and volume calculators to help you find the area and volumes of a range of different 2d and 3d shapes.
Each calculator page comes with worked examples, formulas and practice worksheets.
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How to Print or Save these sheets
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Follow these 3 steps to get your worksheets printed perfectly!
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