# Pentagon

A pentagon is a five-sided polygon. In the figure below are 3 different types of pentagons.

The Pentagon, near Washington, D.C., is one of the world's largest office buildings. From an aerial view, it looks like a pentagon.

## Pentagon classifications

Like other polygons, a pentagon can be classified as regular or irregular. If all the sides and interior angles of a pentagon are equal, it is a regular pentagon. Otherwise it is an irregular pentagon.

Regular pentagon | Irregular pentagon |
---|---|

All sides and interior angles are equal | Not all sides and angles are equal |

Pentagons or other polygons can also be classified as either convex or concave. If all interior angles of a pentagon or polygon are less than 180°, it is convex. If one or more interior angles are larger than 180°, it is concave. A regular pentagon is always a convex pentagon.

Convex pentagon | Concave pentagon |
---|---|

All interior angles < 180° | One or more interior angles > 180° |

## Diagonals of a pentagon

A diagonal is a line segment joining two non-consecutive vertices. Two diagonals can be drawn from each vertex. A total of five diagonals can be drawn for a pentagon. The following figure is an example.

## Internal angles of a pentagon

The sum of the interior angles of a pentagon equals 540°.

As shown in the figure above, two diagonals can be drawn to divide the hexagon into three triangles. The blue lines above show just one way to divide the pentagon into triangles; there are others. The sum of the interior angles of the three triangles equals the sum of interior angles of the pentagon. Since the sum of the interior angles of a triangle is 180°, the sum of the interior angles of the pentagon is 3 × 180° = 540°.

## Regular pentagon

A regular pentagon is a pentagon whose sides are equal in length, and whose interior angles are equal in measure.

- Since each of the interior angles in a regular pentagon are equal in measure, each interior angle measures 540°/5 = 108° as shown below.
- Each exterior angle of a regular pentagon has an equal measure of 72°.

### Symmetry in a regular pentagon

A regular pentagon has 5 lines of symmetry and a rotational symmetry of order 5. This means that it can be rotated in such a way that it will look the same as the original shape 5 times in 360°.

Lines of symmetry | Rotational symmetry |
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5 lines of symmetry | Five 72° angles of rotation |

### Area of a regular pentagon

The area of a regular pentagon with side length s is: