# 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

## Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

## Popular tags

angle area argand diagram cartesian equation chord circle combinations complex polygon complex power complex root cosh cosine cosine rule cube decagon derivative diagonal directrix dodecagon ellipse equilateral triangle eulers formula exponent exponential exterior angle first principles focus gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function infinity interior angle inverse hyperbolic function irregular polygon isosceles trapezium isosceles triangle kite koch curve l system locus major axis minor axis newton raphson method nonagon normal octagon parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root set sine sine rule sinh sloping lines solving equations solving triangles square standard curves star polygon straight line graphs symmetry tangent tanh transformations trapezium triangle turtle graphics vertical