# Decagons - polygons with 10 sides

Categories: gcse geometry

A decagon is a flat shape with 10 straight sides.

## Regular and irregular decagons

A *regular decagon* is a 10-sided shape where every side is the same length and every corner has the same angle. All regular decagons have the same shape, like this:

An *irregular decagon* is a 10 sides shape where not every side and angle are equal. There are many different irregular decagon shapes, here is an example:

## Name

The name *decagon* is a combination of the words *deca* (Greek meaning *five*) and *gonia* (Greek meaning *corner*).

Decagons are sometimes called *10-gons*.

## Interior angles

The interior angles of a decagon are shown here:

The sum of these 10 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 10, so the sum of the interior angles is:

```
(10 - 2) x 180 = 1440 degrees
```

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

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

```
1440 / 10 = 144 degrees
```

## Exterior angles

The exterior angles of a decagon are shown here:

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

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

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

```
360 / 10 = 36 degrees
```

## Symmetry of regular decagons

A regular decagon has 10 lines of symmetry. This means that it can be reflected over each of the 10 grey lines shown here:

A regular decagon has rotational symmetry of order 10. This means that if it rotated about its centre by a 10th of a full turn, it will map onto itself:

## 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
- Octagons - polygons with 8 sides
- Nonagons - polygons with 9 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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