## Polygon

A polygon is a aircraft shape (two-dimensional) with right sides. Examples include triangles, quadrilaterals, pentagons, hexagons and so on.

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## Regular

A "Regular Polygon" has: Otherwise the is | |||

Regular Pentagon | Irregular Pentagon |

Here us look in ~ **Regular Polygons** only.

## Properties

So what can we know around regular polygons? an initial of all, we can work out angles.

## Exterior AngleThe Exterior edge is the edge between any type of side of a shape, |

AlltheExterior angle of a polygon add up come 360°, so:

Each exterior angle need to be 360°/n

(where **n** is the number of sides)

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**Exterior Angle(of a continual octagon)**

### Example: What is the exterior angle of a consistent octagon?

An octagon has 8 sides, so:

## Interior AnglesThe inner Angle and Exterior Angle room measured from the very same line, therefore they add up to 180°. |

Interior angle = 180° − Exterior Angle

We know the** Exterior edge = 360°/n**, so:

Interior edge = 180° − 360°/n

### Example: What is the internal angle that a continuous octagon?

A consistent octagon has actually 8 sides, so:

Exterior edge = 360**° **/ 8 = 45°

internal Angle = 180° − 45° = **135°**

**Interior Angle(of a continuous octagon)**

Or we can use:

### Example: What space the interior and also exterior angle of a regular hexagon?

A regular hexagon has 6 sides, so:

**Exterior edge = 360° **/ 6 = 60°

interior Angle = 180**° − ** 60° = **120°**

And currently for part names:

## "Circumcircle, Incircle, Radius and Apothem ..."

Sounds fairly musical if friend repeat it a couple of times, yet they are just the names of the "outer" and also "inner" circles (and each radius) that deserve to be attracted on a polygon favor this:

**The "outside" circle is dubbed a circumcircle**, and also it connects all vertices (corner points) that the polygon.

The radius the the circumcircle is likewise the **radius** the the polygon.

The "inside" circle is dubbed an **incircle** and also it simply touches each side of the polygon at its midpoint.

The radius that the incircle is the **apothem** the the polygon.

(Not every polygons have actually those properties, however triangles and also regular polygons do).

## Breaking right into Triangles

We can learn a lot about regular polygon by breaking them right into triangles favor this:

Notice that:

the "base" that the triangle is one next of the polygon.the "height" of the triangle is the "Apothem" that the polygonNow, the area the a triangle is fifty percent of the base times height, so:

Area of one triangle = base × elevation / 2 = side × apothem / 2

To gain the area of the whole polygon, just include up the areas of every the small triangles ("n" of them):

Area of Polygon = **n** × next × apothem / 2

And since the perimeter is every the political parties = n × side, we get:

Area the Polygon = perimeter × apothem / 2

## A smaller Triangle

By cut the triangle in half we get this:

**(Note: The angles room in radians, not degrees)**

**The small triangle is right-angled and also so we deserve to use sine, cosine and tangent to uncover how the side**, **radius**, **apothem** and also **n** (number the sides) space related:

sin(π/n) = (Side/2) / Radius | Side = 2 × Radius × sin(π/n) | |

cos(π/n) = Apothem / Radius | Apothem = Radius × cos(π/n) | |

tan(π/n) = (Side/2) / Apothem | Side = 2 × Apothem × tan(π/n) |

There space a lot an ext relationships like those (most of them simply "re-arrangements"), but those will perform for now.

## More Area Formulas

We deserve to use that to calculation the area as soon as we only understand the Apothem:

**Area of little Triangle**= ½ × Apothem × (Side/2)

TypeName when

**RegularSides (n) ShapeInterior AngleRadiusSideApothemArea**

**Equilateral Triangle**

**Square**

**Regular Pentagon**

**Regular**

**Hexagon**

**Regular Heptagon**

**Regular****Octagon**

**...**

**Regular Pentacontagon**Triangle (or Trigon) | 3 | 60° | 1 | 1.732 (√3) | 0.5 | 1.299 (¾√3) | |

Quadrilateral(or Tetragon) | 4 | 90° | 1 | 1.414 (√2) | 0.707 (1/√2) | 2 | |

Pentagon | 5 | 108° | 1 | 1.176 | 0.809 | 2.378 | |

Hexagon | 6 | 120° | 1 | 1 | 0.866 (½√3) | 2.598 ((3/2)√3) | |

Heptagon (or Septagon) | 7 | 128.571° | 1 | 0.868 | 0.901 | 2.736 | |

Octagon | 8 | 135° | 1 | 0.765 | 0.924 | 2.828 (2√2) | |

... | |||||||

Pentacontagon | 50 | 172.8° | 1 | 0.126 | 0.998 | 3.133 | |

(Note: worths correct to 3 decimal places only) |

## Graph

And below is a graph that the table above, however with number of sides ("n") indigenous 3 come 30.

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**Notice that together "n" it s okay bigger, the Apothem is tending towards 1 (equal to the Radius) and that the Area is tending in the direction of π** = 3.14159..., similar to a circle.