## Polygon

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

You are watching: Exterior angle of a regular octagon

## Regular

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

Regular Pentagon | Irregular Pentagon |

Here us look at **Regular Polygons** only.

## Properties

So what can we know around regular polygons? very first of all, we deserve to work out angles.

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

AlltheExterior angles of a polygon include up to 360°, so:

Each exterior angle should be 360°/n

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

Press play button to see.

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

### Example: What is the exterior edge of a continual octagon?

An octagon has 8 sides, so:

## Interior AnglesThe internal Angle and also Exterior Angle room measured indigenous the exact same line, so they add up to 180°. |

Interior edge = 180° − Exterior Angle

We recognize the** Exterior angle = 360°/n**, so:

Interior angle = 180° − 360°/n

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

A constant octagon has actually 8 sides, so:

Exterior edge = 360**° **/ 8 = 45°

inner Angle = 180° − 45° = **135°**

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

Or we can use:

### Example: What room the interior and exterior angles of a consistent hexagon?

A regular hexagon has actually 6 sides, so:

**Exterior angle = 360° **/ 6 = 60°

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

And now for some names:

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

Sounds fairly musical if friend repeat the a few times, however they are simply the surname of the "outer" and also "inner" one (and each radius) that can be drawn on a polygon prefer this:

**The "outside" one is called a circumcircle**, and it connects every vertices (corner points) of the polygon.

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

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

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

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

## Breaking into Triangles

We can learn a lot around regular polygons by break them right into triangles choose this:

Notice that:

the "base" of the triangle is one side of the polygon.the "height" the the triangle is the "Apothem" the the polygonNow, the area of a triangle is half of the basic times height, so:

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

To get the area that the whole polygon, just add up the areas of every the tiny triangles ("n" the them):

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

And because the perimeter is every the sides = n × side, we get:

Area the Polygon = perimeter × apothem / 2

## A smaller sized Triangle

By cutting the triangle in fifty percent we get this:

**(Note: The angles room in radians, no degrees)**

**The tiny triangle is right-angled and also so we deserve to use sine, cosine and tangent to find how the side**, **radius**, **apothem** and **n** (number that sides) room 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 more relationships choose those (most of them simply "re-arrangements"), however those will do for now.

## More Area Formulas

We can use that to calculate the area once we only recognize 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 here is a graph that the table above, yet with variety of sides ("n") native 3 come 30.

See more: Percentage Calculator: What Is 80 Percent Of 150, 80 Percent Of 150

**Notice that together "n" it s okay bigger, the Apothem is tending towards 1 (equal come the Radius) and that the Area is tending towards π** = 3.14159..., as with a circle.