2d Shape With 8 Sides

In geometry, Octagon is a polygon that has viii sides and 8 angles. That ways the number of vertices and edges of an octagon is 8, respectively. In unproblematic words, the octagon is an 8-sided polygon, also chosen 8-gon, in a two-dimensional plane. A regular octagon volition have all its sides equal in length. Each interior bending of a regular octagon is equal to 135°. Therefore, the mensurate of outside angle becomes 180° – 135° = 45°. The sum of the interior angles of the octagon is 135 × 8 = 1080°.In this article, let us discuss the octagon shape, its formulas, properties, and examples in detail.

What is an Octagon?

An octagon is a closed two-dimensional figure with eight sides, eight vertices and eight interior angles. If all the sides and interior angles of an octagon are of equal measure out, then it is chosen a regular octagon otherwise an irregular octagon. The other types of octagons such as convex and concave octagons are also explained in the side by side sections.

Shape of Octagon

Octagon is a geometrical shape in a two-dimensional plane. Similar the other polygon shapes, which we have studied in geometry, suc h as triangle, square, pentagon, hexagon, rectangle, etc., the octagon is also a polygon. The point which defines it as unlike 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 centers of opposite squares form a quadrilateral: equi-diagonal and ortho-diagonal ( whose diagonals length are equal and they bifurcate each other at 90 degrees).

Octagon 1

You lot can meet in the above figure, there are 8 sides of the polygon and eight vertices likewise. This is a regular octagon considering all the angles and sides here are equal.

Real-life Examples of Octagon

Nosotros might have observed that different objects that we apply in our everyday life contain an octagonal shape. Some of the examples include the post-obit:

Outline of an umbrella with 8 ribs
Stop signboard at the signals
A wall clock with 8 edges

Octagon 2

Octagon sides

In the previous section, you learned nearly the shape of an octagon with the help of an image. Permit's understand octagon sides and how to represent them.

How many sides does an octagon accept?

As we already know, the octagon is an 8-sided polygon. That means an octagon contains 8 sides. Also, based on the length of these sides, octagons are classified every bit regular and irregular octagons.

Octagon angles

As defined higher up, the octagonal shape contains 8 angles at viii vertices. Thus, the octagon holds 8 sides and viii angles. There are 8 interior angles and viii exterior angles in an octagon. Octagon interior angles sum is equal to 1080 degrees. Also, the sum of all eight exterior angles is equal to 360 degrees. Based on the type of angles, octagons are classified as convex and concave octagons.

Properties of Octagon

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

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

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 a regular octagon. But if it has unequal sides and unequal angles, it is defined equally an irregular octagon. See the figure below to meet the difference betwixt them.

A regular octagon is a airtight shape with sides of equal length and interior angles of the same measurement. It has eight symmetric lines and a rotational equilibrium of club eight. The interior angle at each vertex of a regular octagon is 135°. The primal angle is 45°.

Octagon 3

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 pointin one thousand inwards, is a convex octagon. The angles of the convex octagon are not more 180°. And the octag on, with on due east of its angles pointi ng in, is a concave-shaped octagon.

Octagon 4

In the above figure, yous can encounter, the convex octagon has all its angles pointing exterior from the center or origin point. Whereas on the right side, the concave octagon has one of the angles pointing

towards the inside of the polyg on.

Diagonals of Octagon

For any n-sided polygon, we tin can find the number of diagonals using the formula n(n – 3)/ii.

Similarly, we tin can find the number of diagonals in an octagon.

For octagon, n = 8

Substituting n – 8 in the required formula, we go;

n(n – three)/2 = viii(eight – 3)/2 = iv(5) = 20

Therefore, an octagon contains a total of 20 diagonals. These can be fatigued as shown in the below figure.
Octagon 5

Length of the Diagonal of Octagon

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

Fifty = a√(four + 2√two)

where a is the side of the octagon.

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.

Area of Regular Octagon

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

Expanse = 2a²(1 + √2)
This is the octagon area formula in case of equal sides.

Octagon Lines of symmetry

Nosotros know that the line of symmetry of a shape divides it into identical halves. Thus, beingness an 8 sided polygon, it has 8 lines of symmetry. However, these lines of symmetry can be drawn for regular octagons and this tin can be shown as:
Octagon 6

Octagonal Prism

An octagonal prism has two octagonal faces and 8 foursquare sides. The shape of an octagonal prism is given equally:
Octagon 7

Octagonal Pyramid

If the base of a pyramid is an octagon, then nosotros call information technology an octagonal pyramid. The shape of an octagonal pyramid is given in the figure below.
Octagon 8

Area of Regular and Irregular Octagon
Area of an Octagon Formula
Octagon Formula For Area and Perimeter
Octagon Calculator
Area of An Octagon Computer

Solved Examples on Octagon

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

Solution:

Given, a = 5 cm

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

And Expanse = 2aⁱⁱ(ane+√2) = two × 5² (1+√2) = 2 × 25 (1+√2)= 120.vii cm²

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

Solution:

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

Expanse = 2aⁱⁱ(1+√2) = 2 (vii)^two(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 x cm.

Solution:

Given,

Length of side of octagon = a = x cm

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

L = a√(4 + two√two)

Hence,

L = 10√(4 + ii√ii)

L = x ten √6.828

L= 10 x ii.613

L = 26.13 cm

Q.4: Observe the area and perimeter of a regular octagon whose side is of length two.5 cm.

Solution:

Given,

Length of side of octagon = a = ii.5 cm

Surface area of octagon = 2a^two(ane+√2)

A = ii x (two.5)²(1+√2)

A = 12.5 x (1+√2)

A = 30.177 sq.cm

Perimeter of Octagon = 8 10 sides of the octagon

P = viii x 2.v

P = 20cm

Frequently Asked Questions on Octagon

What is the shape of the octagon?

In geometry, an octagon is a polygon that has 8 sides and eight angles. All the sides are joined end to cease to course the shape of the octagon. The sum of the interior angles of an octagon is equal to 1080 degrees.

What are the different classifications of the octagon?

The octagon can exist classified into:
Regular and irregular octagon
Convex and concave octagon

What is meant by the regular and irregular octagon?

If all the sides and angles of an octagon are of the same measure, and then information technology is chosen the regular octagon. If the octagon has unequal side length and bending, and so information technology is called an irregular octagon.

Ascertain convex and concave octagon.

If all the angles of an octagon are pointing outwards and the interior angles are non more than than 180 degrees, then it is chosen a convex polygon. If any of the angles are pointing in wards, then it is called a concave polygon.

Mention the properties of the octagon.

The important properties of the octagon are:
Octagon has 8 sides and 8 angles
The sum of the interior angles of an octagon is equal to 1080 degrees.
There are twenty diagonals in a regular octagon.

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