# Pentagons - polygons with 5 sides

Categories: gcse geometry

A pentagon is a flat shape with 5 straight sides.

## Regular and irregular pentagons

A *regular pentagon* is a 5-sided shape where every side is the same length and every corner has the same angle. All regular pentagons have the same shape, like this:

An *irregular pentagon* is a 5 sides shape where not every side and angle are equal. There are many different irregular pentagon shapes, here is an example:

## Name

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

Pentagons are sometimes called *5-gons*.

## Interior angles

The interior angles of a pentagon are shown here:

The sum of these 5 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 5, so the sum of the interior angles is:

```
(5 - 2) x 180 = 540 degrees
```

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

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

```
540 / 5 = 108 degrees
```

## Exterior angles

The exterior angles of a pentagon are shown here:

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

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

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

```
360 / 5 = 72 degrees
```

## Symmetry of regular pentagons

A regular pentagon has 5 lines of symmetry. This means that it can be reflected over each of the 5 grey lines shown here:

A regular pentagon has rotational symmetry of order 5. This means that if it rotated about its centre by a 5th of a full turn, it will map onto itself:

## Real life examples

Pentagons appear quite often in nature. If you slice an apple in half horizontally and look at the cross section of its core, it forms a 5 pointed star whose points make a pentagon.

## See also

- Regular polygons
- Interior and exterior angles of a polygon
- Triangles
- Quadrilaterals
- Quadrilateral family tree
- Hexagons - polygons with 6 sides
- Heptagons - polygons with 7 sides
- Octagons - polygons with 8 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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