If you are interesting in learning how to calculate the area, perimeter, apothem or circumcircle radius of a regular polygon, check out our Polygon Calculator.

For more tools related to geometry, triangle, and generally polygons and their properties, check out the list below:

What is a Polygon?

According to the polygon definition, a polygon is a flat 2D geometric shape closed by straight line segments. It has no limitation in terms of its sides count. But, each needs to be straight, not curved.

Polygon is a flat 2D geometric shape closed by straight line segments.

The term polygon comes from the Greek word “polugonos”, but it evolved into “polygon” over the years.

The main classification of polygons groups them into two categories:

  • Regular polygons – all sides are of the same size and angle.
  • Irregular polygons – all sides and/or angle of different lengths.

Another classification classifies polygons into:

  • Simple polygons – shapes with a single boundary and no intersecting lines. Mostly simple polygons are generally called only “polygons” omitting the word “simple”.
  • Complex polygon – more than one boundary, and they can have intersecting sides.

Polygons can also be concave or convex:

  • Concave polygons – shapes with internal angles greater than 180° and pointing inwards.
  • Convex polygons – shapes with no angles pointing inwards, meaning all of them are less than 180°.

Types of Polygon

Depending on the number of sides a polygon is made of, we have different types of polygons. Therefore, in the table below, you can see a list of polygons, their names, sides count for each, and angles.

Namen (sides)αα
3 – sided polygontriangleπ/3 = 60°2π/3 = 120°
4 – sided polygontetragonπ/2 = 90°π/2 = 90°
5 – sided polygonpentagon3π/5 = 108°2π/5 = 72°
6 – sided polygonhexagon2π/3 = 120°π/3 = 60°
7 – sided polygonheptagon5π/7 = 128.57°2π/7 = 51.43°
8 – sided polygonoctagon3π/4 = 135°π/4 = 45°
9 – sided polygonnonagon7π/9 = 140°2π/9 = 40°
10 – sided polygondecagon8π/10 = 144°π/5 = 36°
n – sided polygonn – gon(n-2) * 180°/n360°/n
Types of Polygon

Polygons with (n = 10+) are less common and popular, but still, they have their names. You can learn more about it in the additional table below:

PolygonNamen (sides)αβ
11 – sided polygonHendecagon11147.273°32.73°
12 – sided polygonDodecagon12150°30°
13 – sided polygonTriskaidecagon13152.308°27.69°
14 – sided polygonTetrakaidecagon14154.286°25.71°
15 – sided polygonPentadecagon15156°24°
16 – sided polygonHexakaidecagon16157.5°22.5°
17 – sided polygonHeptadecagon17158.824°21.18°
18 – sided polygonOctakaidecagon18160°20°
19 – sided polygonEnneadecagon19161.053°18.98°
20 – sided polygonIcosagon20162°18°
30 – sided polygonTriacontagon30168°12°
40 – sided polygonTetracontagon40171°
50 – sided polygonPentacontagon50172.8°7.2°
60 – sided polygonHexacontagon60174°
70 – sided polygonHeptacontagon70174.857°5.14°
80 – sided polygonOctacontagon80175.5°4.5°
90 – sided polygonEnneacontagon90176°
Types of Polygon (n+10)

Polygon Formulas

Our calculator and all the formulas listed below are only applicable if we talk about regular polygons. Therefore, if you need to find irregular polygons‘ area, perimeter, or other aspects, you shouldn’t refer to those formulas.

So, for calculating the properties of a regular polygon in geometry, you only need to know its side length and shape.

Area of Polygon

\text{A} = n \times a² \times \cot \frac {(π/n)} {4}

where: 

n – number of sides
a – side length

We also have a specially developed a tool for calculating the area of a regular polygon, so check it out.

Perimeter of Polygon

\text{P} = n \times a

For a more detailed approach in calculating the perimeter of a regular polygon, you can use our Perimeter of a Polygon.

Angles

Each regular polygon has interior and exterior angles. So, what about them? What’s the formula for finding them?

Interior angle of a polygon

α = (n - 2) \times \frac {π} {n}

where:

n – number of sides
α – an interior angle

Exterior angle of a polygon

β = 2 \times \frac {π} {n}

where:

β – an exterior angle
n – number of sides

There is our useful Angle Conversion tool which you should check out.

Incircle radius (Apothem)

Apothem or incircle radius of a regular polygon can be calculated by only knowing the side length and the number of sides of a regular polygon.

\text{Apothem} = \frac {a}  {2 \times tan(π/n)}

where:

a – side length
n – number of sides

Circumcircle radius

What about finding the circle that passes through every vertex of a regular polygon? Well, you can get it by using the formula shown below:

\text{Circumcircle radius} = \frac {a}  {2 \times sin(π/n)}

Polygon Calculator – How to Use?

If you ever need a tool that can help you measure every aspect of regular polygons, check out our Polygon Calculator. Why? Because it is an all-in-one tool that allows you to find the area, perimeter, apothem or circumcircle radius of polygon just by entering the sides count and the side length of a polygon.

Steps:

  • Head over to our calculator (web or mobile version) and enter the number of sides of a polygon that you want to find
  • It will also, based on what you input, display the name of that polygon if it finds any (e.g., 5 -> pentagon)
  • After choosing the polygon, you must specify the side length (a), and the calculator will calculate all the aspects. For example, this is very handy because by knowing only one parameter, you can get the rest. 
  • The calculator returns and displays the result in a unit that you specify. For instance, you can measure in centimeters, meters, inches, feet, yards, etc.

Polygon Calculator – Example

In the “How to Use” section, we have shown you and explained the process of using our calculator to help you get all the aspects of a regular polygon. Therefore, in this section, we will see how to utilize the full power of it in a real example below:

Scenario: How to find the area, perimeter, apothem and circumcircle radius of a seven-sided polygon (septagon)? The side length of the given polygon is 8 cm.

Steps:

  1. Set “number of sides” to 7
  2. Enter “7” in the area (a) parameter field
  3. The Polygon Calculator uses the formula from above and returns the result:

Side (a): 8 cm

P: 56 cm

A: 232.57 cm²

α: 128.57 deg

β: 51.43 deg

Circumradius (R): 9.219 cm

Apothem (r): 8.306 cm

FAQ

How many sides does a polygon have?

There are some standard polygon names when the number does not exceed 20. However, it doesn’t mean that there cannot be polygons with 20+ sides. For example:
– three-sided is called a triangle
– four-sided is called a square
– eleven-sided is undecagon
– 50-sided is 55-gon 
– and so on.

How to calculate the area of a pentagon?

You can find the area of a pentagon (five-sided polygon) using the standard polygon’s area formula below:
Area = n \times a² \times \cot(π/n)/4
Note: For pentagons, n = 5.

What is the sum of interior angles of a polygon?

The sum of interior angles of a polygons equals 180 degrees.

Is a circle a polygon?

For a shape to be a polygon, it needs to be composed of finite line segments that are straight, not curved. For example, a circle is not a shape made of straight lines but a geometric figure closed by curved line segments; thus, we can’t call it a polygon.

What is a polygon with four sides?

A four-sided polygon is a square.

What polygon has nine sides?

A nonagon is a polygon that has nine sides.