# Octagons - polygons with 8 sides

Categories: gcse geometry

A octagon is a flat shape with 8 straight sides.

## Regular and irregular octagons

A *regular octagon* is a 8-sided shape where every side is the same length and every corner has the same angle. All regular octagons have the same shape, like this:

An *irregular octagon* is a 8 sides shape where not every side and angle are equal. There are many different irregular octagon shapes, here is an example:

## Name

The name *octagon* is from the Greek *octagonon* (meaning *octagon*).

Octagons are sometimes called *8-gons*.

## Interior angles

The interior angles of a octagon are shown here:

The sum of these 8 angles is given by the formula:

```
sum of interior angles = (n - 2) x 180
```

Where *n* is the number of sides. In this case, the number of sides *n* is 8, so the sum of the interior angles is:

```
(8 - 2) x 180 = 1080 degrees
```

For a regular octagon, all the interior angles are equal:

This means that the interior angle of a regular octagon is:

```
1080 / 8 = 135 degrees
```

## Exterior angles

The exterior angles of a octagon are shown here:

The sum of the exterior angles of *any polygon* is 360 degrees.

For a regular octagon, all the interior angles are equal:

This means that the exterior angle of a regular octagon is:

```
360 / 8 = 45 degrees
```

## Symmetry of regular octagons

A regular octagon has 8 lines of symmetry. This means that it can be reflected over each of the 8 grey lines shown here:

A regular octagon has rotational symmetry of order 8. This means that if it rotated about its centre by a 8th of a full turn, it will map onto itself:

## Real life examples

Octagons are common design elements: signs (traffic Stop signs are octagonal in many countries), floor tiles, biscuit tins, even quirky window frames. This is partly because the angles in a regular octagon are all multiples of 45 degrees which often makes items of that shape easier to manufacture.

## See also

- Regular polygons
- Interior and exterior angles of a polygon
- Triangles
- Quadrilaterals
- Quadrilateral family tree
- Pentagons - polygons with 5 sides
- Hexagons - polygons with 6 sides
- Heptagons - polygons with 7 sides
- Nonagons - polygons with 9 sides
- Decagons - polygons with 10 sides
- Hendecagons - polygons with 11 sides
- Dodecagons - polygons with 12 sides
- n-gons - polygons with any number of sides
- Other types of polygon
- Star polygons

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