# Pentagon

A pentagon is a 5-sided Polygon (a flat shape with straight sides):

## Regular or Irregular

When all angles are equal and all sides are equal it is **regular**, otherwise it is** irregular**:

Regular Pentagon | Irregular Pentagons |

## Concave or Convex

A **convex** pentagon has no angles pointing inwards. More precisely, no internal angles can be more than 180°.

When any internal angle is greater than 180° it is **concave**. (*Think: concave has a "cave" in it*)

Convex Pentagon | Concave Pentagon |

## Is it a Pentagon?

A pentagon has 5 **straight** sides. The shape must also be **closed** (all the lines connect up):

Pentagon
(straight sides) |
Not a Pentagon
(has a curve) |
Not a Pentagon
(open, not closed) |

## Properties

A **regular** pentagon has:

- Interior Angles of
**108°** - Exterior Angles of
**72°** - Area of approximately
**1.7204774 × s**(where s=side length)^{2}

**Any** pentagon has:

- Sum of Interior Angles of
**540°** - 5 diagonals

## Make a Regular Pentagon

You can make a regular pentagon with a strip of paper!

Start with a long strip of paper, make sure it is the same width all along (if you want the pentagon to be regular):

Make a "pretzel" knot with the paper.

Carefully tighten the knot while keeping the paper flat.

.....

Trim off (or fold back) any excess.

All sides are now of equal length, and all angles should be the same too.

## Images

This Petunia is almost a perfect pentagon

The "**Pentagon**" in Washington DC

is 5 regular pentagons with a common center.

## Masterclass

How to construct a Pentagon using just a compass and a straightedge.