We have spoken before about polygons, but we have not tackled regular polygons by themselves. They are the crème de la crème of polygons, a class unto themselves!

## What is a Polygon?

Polygons are straight-sided, flat shapes that close in a space. This means every polygon:

• Is two-dimensional
• Uses only line segments to create its sides
• Has an interior and exterior

The word "polygon" comes from the Greek and means "many angles," because all simple polygons have as many sides as they have angles. The simplest polygon, the triangle, has three interior angles (and three exterior angles!) so it also has three sides.

You can have polygons with so many sides, they appear to be circles even when they are not. Once you get above 10 sides, most mathematics students just say "n-gon," where n is the number of sides or interior angles, as in a "18-gon," or a "25-gon," rather than trying to remember the Greek-origin name.

## What is a Regular Polygon?

To be a regular polygon, the flat, closed, straight-sided shape must also have another property. Every interior and exterior angle in the regular polygon is equal to every other interior and exterior angle, and every side is equal in length to every other side.

A regular polygon has:

• Two dimensions
• Straight sides
• Congruent (equal-length) sides
• An interior and exterior
• Equal interior angles
• Equal exterior angles

### Regular Polygon Examples

A few examples of regular polygons are a triangle, quadrilateral, pentagon, hexagon, heptagon, and a decagon.

### Regular Vs. Irregular Polygons

If even one of those features is not present, you do not have a regular polygon. You may have three of the features (two dimensions; straight sides; an interior and exterior) but still not have a regular polygon. You would have an irregular polygon.

A rectangle that is longer than it is tall is an example of an irregular polygon. A scalene triangle, home plate on a baseball or softball field, and a kite are all also examples of irregular polygons.

## Properties of Regular Polygons

Only when all six conditions, outlined above, are present will you have a regular polygon. Let's examine a suspicious shape and see:

Here we have $△CAT$, drawn with black lines. So our black $CAT$ is suspicious, indeed.

Go through the six properties of regular polygons:

• Is it two-dimensional?
• Is it made with straight sides?
• Are its sides congruent?
• Does it close in a space, creating an interior and exterior?
• Does it have equal interior angles?
• Does it have equal exterior angles?

Results of our scrutiny using the six identifying properties of regular polygons: $△CAT$ is a regular polygon, the simplest kind. It is an equilateral triangle, with interior angles of $60°$ and exterior angles of $120°$.

## Names of Regular Polygons

Imagine you are unpacking a shipment of polygons. You have to put all the irregular polygons on the discount shelf, and all the regular polygons on the full-price shelf. Using your knowledge of the identifying properties of regular polygons, you can easily see that only a few typical polygons are regular:

Regular Polygon Number of Sides Interior Angles Exterior Angles
Equilateral triangle 3 sides 3 interior angles of 60° 3 exterior angles of 120°
Square 4 sides 4 interior angles of 90° 4 exterior angles of 90°
Regular pentagon 5 sides 5 interior angles of 108° 5 exterior angles of 72°
Regular hexagon 6 sides 6 interior angles of 120° 6 exterior angles of 60°
Regular heptagon 7 sides 7 interior angles $\cong$ 128.57° 7 exterior angles $\cong$ 51.43°
Regular octagon 8 sides 8 interior angles of 135° 8 exterior angles of 45°

### How many sides does a regular polygon have?

You can create regular polygons of any number of sides and interior angles, but usually you will not name them with anything other than the number of their sides or interior angles and then -gon:

Regular Polygon Number of Sides Interior Angles Exterior Angles
Regular enneagon or 9-gon 9 sides 9 interior angles of 140° 9 exterior angles of 40°
Regular decagon or 10-gon 10 sides 10 interior angles of 144° 10 exterior angles of 36°
Regular hectogon or 100-gon 100 sides 100 interior angles of 176.4° 100 exterior angles of 3.6°

Any polygon that does not have equal-length sides along with equal interior and exterior angles is an irregular polygon; it goes on the discount shelf!

## Parts Of Regular Polygons

As simple as they are, regular polygons still have six parts:

1. Sides
2. Interior and exterior
3. Interior angles
4. Exterior angles
5. Vertices
6. Diagonals

We will illustrate with a basic regular polygon, the hexagon. This is a familiar shape (think of the cells of a honeybee hive):

Our hexagon has six congruent sides. It has six equal interior angles, each $120°$. It also has six equal exterior angles. An exterior angle is created by extending one side and measuring the angle between that extension and an adjacent side.

## Regular Polygon Angles

### Exterior Angles of a Regular Polygon

Exterior angles of every simple polygon add up to $360°$, because a trip around the polygon completes a rotation, or return to your starting place. Where sides meet, they form vertices, so our hexagon also has six vertices.

### Interior Angles of a Regular Polygon

Inside the hexagon's sides, where the interior angles are, is the hexagon's interior. Outside its sides is the hexagon's exterior. This becomes important when you consider complex polygons, like a star-shape (a pentagram, for example).

### Diagonal of a Polygon

Connect any non-adjacent vertices in the interior of the hexagon to make diagonals. All simple polygons can be divided into triangles using diagonals. The minimum number of triangles created in our hexagon, by drawing three diagonals, is four; each triangle has interior angles adding to $180°$, so the sum of the interior angles of a regular hexagon is , or $720°$.

## Regular Polygon Formula

The formula for calculating the sum of interior angles of any regular polygon with $n$ sides is:

To find any single interior angle, divide your answer by $n$.

Need more help? Here's a step by step lesson and formulas to finding interior and exterior angles of polygons.

## Lesson Summary

Now you have all the tools you need to identify polygons, tell regular ones from irregular ones, name regular polygons and recognize them by their properties, list the parts of regular polygons (sides, interior, interior angles, vertices, diagonals, exterior angles), and calculate the sums of the interior angles of polygons.

## What you learned:

After viewing the video and studying the drawings, please read the entire lesson. Then you will learn to:

• Identify polygons
• Identify regular and irregular polygons
• State the identifying properties of regular polygons
• Name examples of regular polygons
• List six parts of regular polygons
• Calculate the sum of interior angles of regular polygons
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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