 # Octagon

Octagon is a polygon in geometry, which has 8 sides and 8 angles. That means the number of vertices is 8 and the number of edges is 8. All the sides are joint with each other end-to-end to form a shape. These sides are in a straight line form, they are not curved or disjoint with each other. Each interior angle of a regular polygon is 135°, therefore, the measure of exterior angle becomes 180° – 135° = 45°. The sum of the interior angles of the octagon is 135 × 8 = 1080°. You can see in the above figure, there are 8 sides of the polygon and have eight vertices as well. This is a regular octagon because all the angles and sides here are equal.  In the same way, based on sides and angles there are many types of polygons such as:

• Triangle
• Pentagon
• Hexagon
• Septagon
• Nonagon
• Decagon and so on.

## Shape of Octagon

Octagon is a geometrical shape in a two-dimensional plane. Like the other polygon shapes, we have studied in geometry, such as triangle, square, pentagon, hexagon, rectangle, etc., the octagon is also a polygon. The points which define it different from other geometrical shapes is that it has 8 sides and 8 angles.

If squares are built internally or externally on all the sides of an octagon, then the midpoints of the sections joining the centres of opposite squares form a quadrilateral: equi-diagonal and orthodiagonal ( whose diagonals length are equal and they bisect each other at 90 degrees).

## Types of Octagon

Depending upon the sides and angles of the octagon, it is classified into the following categories;

• Regular and Irregular Octagon
• Concave and Convex Octagon

### Regular and Irregular Octagon

When an octagon has all equal sides and equal angles, then it is defined as regular octagon but with unequal sides and unequal angles, it is defined as an irregular octagon. See the figure below to see the difference between them.

A regular octagon is a closed shape with sides of the equal length and interior angles of the same measurement. It has eight symmetric lines and rotational equilibrium of order 8. The interior angle at each vertex of a this octagon is 135°. The central angle is 45°. In the above figure, the left-hand side figure depicts a regular octagon and the two figures on the right side shows irregular octagons. From the figure itself, we can analyse that there is a difference between the symmetry of regular and irregular polygons.

### Convex and Concave Octagon

The octagon which has all its angles pointing outside or no angles are pointing inwards, is a convex octagon. The angles of the convex octagon are not more than 180°. And the octagon, which one of its angles pointing inward is a concave-shaped octagon. In the above figure, you can see, the convex octagon has all its angles pointing outside from the center or origin point. Whereas on the right side, the concave octagon has one of the angles pointing towards inside the polygon.

## Properties of Octagon

In the case of properties, we usually consider regular octagons.

• These have eight sides and eight angles.
• All the sides and all the angles are equal respectively.
• There is a total of 20 diagonals in a regular octagon.
• The total sum of the interior angles is 1080°, where each angle is equal to 135°(135×8 = 1080)
• Sum of all the exterior angles of the octagon is 360° and each angle is 45°(45×8=360).

### Area of Regular Octagon

Area of the octagon is the region covered by the sides of the octagon. The formula for area of a regular octagon which has 8 equal sides and all its interior angles are equal to 135°, is given by:

Area = 2a2(1+√2)

### Perimeter of Octagon

The perimeter of the octagon is the length of the sides or boundaries of the octagon, which forms a closed shape.

Therefore,

Perimeter = Sum of all Sides = 8a

Where a is the length of one side of the octagon.

### Length of the Diagonal of Octagon

If we join the opposite vertices of a regular octagon, then the diagonals formed have the length equal to:

L = a√(4 + 2√2)

where a is the side of the octagon.

### Octagon Example

Q.1: If the length of the side of a regular octagon is 5cm. Find its perimeter and area.

Solution: Given, a = 5cm

Therefore, Perimeter = 8a = 8 × 5 = 40cm

And Area = 2a2(1+√2) = 2 × 52 (1+√2) = 2 × 25 (1+√2)= 120.7 cm2

Q.2: If the side length of a regular octagon is 7cm. Find its area.

Solution: Given, length of the side of the octagon, a = 7cm

Area = 2a2(1+√2) = 2 (7)2(1+√2) =  236.6 sq.cm.

Q.3: Find the length of the longest diagonal of a regular octagon whose side length is equal to 10cm.

Solution: By the formula, we know, the length of the longest diagonal formula is given by:

L = a√(4 + 2√2)

Hence,

L = 10√(4 + 2√2)

L = 10 x 2.613

L = 26.13 cm

Q.4: Find the area and perimeter of a regular octagon whose side is of length 2.5 cm.

Solution: Area of octagon = 2a2(1+√2)

A = 2 x (2.5)(1+√2)

A = 12.5 x (1+√2)

A = 30.177 sq.cm

Perimeter of Octagon = 8 x sides of the octagon

P = 8 x 2.5

P = 20cm

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