# Octagon - Shape, Types, Properties, Angle sum property, perimeter of an octagon and Area of Octagon

In geometry, a shape formed by joining straight lines is called a polygon. A polygon can have any number of sides, the smallest being a triangle which is a three-sided polygon. Polygons are geometric shapes named on the basis of a number of sides. For example,

1. Triangle- a three-sided polygon
2. Quadrilateral- a four-sided polygon. Examples: square, rectangle, parallelogram
3. Pentagon- a five-sided polygon
4. Hexagon- a six-sided polygon
5. Heptagon- a seven-sided polygon
6. Octagon- an eight-sided polygon

And so on, depending on the number of sides and interior angles. In Geometry circle is considered a polygon that has infinite sides.

## Octagon

An octagon is an eight-sided closed polygon. It has eight vertices which make eight interior angles.

The above is an octagon sides AB, BC, CD, DE, EF, FG, GH and HA.

## Types of octagons

Octagons can be classified on the basis of length of sides and angles as

1. Regular octagon - A regular octagon is one that has all its sides equal. The length of the sides of a regular octagon is the same. The measure of each of its interior angles is 135°. This also means that each exterior angle corresponding to an interior angle in a regular heptagon is 45.
Interior angle = 135°
Exterior angle = 180° – interior angle
Exterior angle = 180° – 135°
Exterior angle = 45°
2. Irregular octagon - An irregular octagon is one that has all its sides and angles different from each other. Sides are not equal in length, and angles are unequal in measurement.
3. Convex octagon - An octagon which has all its angles directed outwards is called a convex octagon. All the angles in a convex octagon are less than 180°.
4. Concave octagon - An octagon which has any of its angles directed inwards is called a concave octagon. The angles in a concave octagon are more than 180°.

## Properties of an octagon

1. For A regular Octagon, all the sides and angles are equal.
2. An octagon can have 20 diagonals.
3. The sum of its interior angles is equal to 1080° in a regular octagon.
4. For a regular octagon, the sum of its exterior angles is 360°.

## Angle sum property of a Octagon

The angle sum property of an octagon is the sum of all its interior angles. It can be calculated by the given formula.
(n – 2) . 180°
Angle sum property of an octagon = (8 – 2). 180°
Angle sum property of an octagon = 6 . 180°
Angle sum property of an octagon = 1080°
Where n is the number of sides.

## The perimeter of an octagon

The perimeter of an octagon is the length of the boundary of an octagon. It can be calculated by adding all sides of a given octagon. For a regular octagon, the perimeter can be calculated by the formula.

Perimeter = 8 . length of side

For an irregular octagon, the perimeter is the sum of it’s sides.

## Area of an octagon

The perimeter of a heptagon is the measure of the total boundary length of the heptagon. It is calculated by adding all the sides of the given heptagon. For a regular heptagon, the perimeter of the heptagon is obtained as the product of measure of one of its sides by 7.

Perimeter of a regular heptagon = 7 . length of a side

## Area of Octagon

The area of an octagon can be obtained by the formula as below:

Area = 2(1 + √2) . side²