Polygons are everywhere! In this lesson, you will learn what they are and what they look like. You will also learn about a special class of polygons and how to find its angles.

## Definition of a Polygon

A **polygon** is any 2-dimensional shape formed with straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. The name tells you how many sides the shape has.

For example, a triangle has three sides, and a quadrilateral has four sides. So, any shape that can be drawn by connecting three straight lines is called a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral.

Shape | # of Sides |
---|---|

Triangle | 3 |

Square | 4 |

Rectangle | 4 |

Quadrilateral | 4 |

Pentagon | 5 |

Hexagon | 6 |

Heptagon | 7 |

Octagon | 8 |

Nonagon | 9 |

Decagon | 10 |

n-gon |
n sides |

The last entry includes the general term for a polygon with *n* number of sides. Polygons aren’t limited to the common ones we know but can get pretty complex and have as many sides as are needed. They can have 4 sides, 44 sides, or even 444 sides.

The names would be 4-gon, or quadrilateral, 44-gon, and 444-gon, respectively. An 11-sided shape can be called an 11-gon.

## Regular Polygons

A special class of polygon exists; it happens for polygons whose sides are all the same length and whose angles are all the same. When this happens, the polygons are called **regular polygons**. A stop sign is an example of a regular polygon with eight sides.

All the sides are the same and no matter how you lay it down, it will look the same. You wouldn’t be able to tell which way was up because all the sides are the same and all the angles are the same.When a triangle has all the sides and angles the same, we know it as an equilateral triangle, or a regular triangle.

A quadrilateral with all sides and angles the same is known as a square, or regular quadrilateral. A pentagon with all sides and angles the same is called a regular pentagon. An *n*-gon with sides and angles the same is called a regular *n*-gon.

Here is a regular triangle, a regular quadrilateral, and a regular pentagon. Do you see how all the sides are the same and no matter how you flip it, it will look the same?

## Angles of Regular Polygons

Regular polygons also have two different angles related to them.

The first is called the **exterior angle**, and it is the measurement between the shape and each line segment when you stretch it out past the shape.

However many sides a polygon has is the same number of exterior angles it has. So, a pentagon with five sides has five exterior angles.

A hexagon will have six exterior angles and so on. For regular polygons, we can figure out the measurement of the exterior angle, but for polygons that aren’t regular, we can’t. Here is the formula for regular polygons:

You can also say that the interior angle is the measurement of each corner of the polygon.

Here is the formula for the interior angle:

The second formula is the same as the first, just rearranged. Don’t worry about how we got there right now; just remember one or the other, and you will be okay. The second one is the more commonly seen in the math world.

Let’s look at an example. For our pentagon with five sides, using the first equation gives us 180 – 360 / 5 = 180 – 72 = 108 degrees. Using the second equation, we get (5 – 2) * 180 / 5 = 3 * 180 / 5 = 540 / 5 = 108 degrees. Both formulas will give us the same answer. Choose the formula that is easier for you to remember.

## Lesson Summary

Polygons are all around us.

Who of us has ever seen a triangle or a square? A **polygon** is defined as a 2-dimensional shape with straight sides. **Regular polygons** have sides and angles that are all the same. While you can find the measurements of the exterior and interior angles of regular polygons, you can’t with polygons that aren’t regular.

## Learning Outcomes

After this lesson, you should be able to:

- Define polygon and regular polygon
- Identify examples of polygons and regular polygons
- Explain how to find the exterior and interior angles of regular polygons