Interior Angles Of A Octagon

In geometry, Octagon is a polygon that has 8 sides and 8 angles. That ways the number of vertices and edges of an octagon is viii, respectively. In simple words, the octagon is an 8-sided polygon, likewise chosen 8-gon, in a ii-dimensional plane. A regular octagon volition accept all its sides equal in length. Each interior angle of a regular octagon is equal to 135°. Therefore, the measure of exterior angle becomes 180° – 135° = 45°. The sum of the interior angles of the octagon is 135 × 8 = 1080°.In this article, permit us discuss the octagon shape, its formulas, backdrop, and examples in item.

What is an Octagon?

An octagon is a airtight two-dimensional effigy with eight sides, viii vertices and viii interior angles. If all the sides and interior angles of an octagon are of equal measure, then information technology is called 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 2-dimensional airplane. Like the other polygon shapes, which we take studied in geometry, suc h as triangle, square, pentagon, hexagon, rectangle, etc., the octagon is also a polygon. The bespeak which defines it every bit different from other geometrical shapes is that information technology 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 grade a quadrilateral: equi-diagonal and ortho-diagonal ( whose diagonals length are equal and they bifurcate each other at 90 degrees).

Octagon 1

Y'all tin can see in the above effigy, at that place are eight sides of the polygon and eight vertices equally well. This is a regular octagon considering all the angles and sides here are equal.

Existent-life Examples of Octagon

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

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

Octagon 2

Octagon sides

In the previous section, you learned near the shape of an octagon with the assist of an image. Let's understand octagon sides and how to correspond them.

How many sides does an octagon have?

As nosotros already know, the octagon is an viii-sided polygon. That ways an octagon contains viii sides. Also, based on the length of these sides, octagons are classified as regular and irregular octagons.

Octagon angles

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

Properties of Octagon

In the case of backdrop, we commonly consider regular octagons.

These accept eight sides and eight angles.
All the sides and all the angles are equal, respectively.
There are 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 outside 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 post-obit categories;

Regular and Irregular Octagon
Concave and Convex Octagon

Regular and Irregular Octagon

When an octagon has all equal sides and equal angles, and then it is divers as a regular octagon. But if information technology has unequal sides and unequal angles, it is divers as an irregular octagon. Meet the figure below to meet the difference between them.

A regular octagon is a closed shape with sides of equal length and interior angles of the same measurement. It has eight symmetric lines and a rotational equilibrium of order 8. The interior bending at each vertex of a regular octagon is 135°. The central angle is 45°.

Octagon 3

In the higher up effigy, the left-hand side effigy 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 g inward, is a convex octagon. The angles of the convex octagon are not more 180°. And the octag on, with on e of its angles pointi ng in, is a concave-shaped octagon.

Octagon 4

In the in a higher place effigy, you can run into, 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 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)/2.

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

For octagon, n = viii

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

northward(northward – three)/2 = eight(8 – 3)/2 = 4(5) = twenty

Therefore, an octagon contains a full of xx diagonals. These tin can exist fatigued as shown in the beneath figure.
Octagon 5

Length of the Diagonal of Octagon

If we bring together the contrary vertices of a regular octagon, and so the diagonals formed have the length equal to:

L = a√(four + two√2)

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.

Surface area of Regular Octagon

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

Area = 2a²(ane + √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, existence an eight sided polygon, it has viii lines of symmetry. However, these lines of symmetry can be drawn for regular octagons and this can be shown as:
Octagon 6

Octagonal Prism

An octagonal prism has ii octagonal faces and 8 square sides. The shape of an octagonal prism is given as:
Octagon 7

Octagonal Pyramid

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

Surface area of Regular and Irregular Octagon
Area of an Octagon Formula
Octagon Formula For Expanse and Perimeter
Octagon Calculator
Surface area of An Octagon Estimator

Solved Examples on Octagon

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

Solution:

Given, a = 5 cm

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

And Area = 2a^two(ane+√ii) = 2 × 5² (i+√2) = 2 × 25 (1+√two)= 120.7 cmⁱⁱ

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

Solution:

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

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

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

Solution:

Given,

Length of side of octagon = a = ten cm

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 √6.828

L= 10 x 2.613

L = 26.13 cm

Q.iv: Notice the area and perimeter of a regular octagon whose side is of length two.v cm.

Solution:

Given,

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

Area of octagon = 2a^two(1+√2)

A = 2 x (2.5)²(i+√two)

A = 12.5 x (ane+√2)

A = 30.177 sq.cm

Perimeter of Octagon = 8 x sides of the octagon

P = 8 x two.5

P = 20cm

Often Asked Questions on Octagon

What is the shape of the octagon?

In geometry, an octagon is a polygon that has 8 sides and 8 angles. All the sides are joined end to cease to grade 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 be 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 mensurate, then it is called the regular octagon. If the octagon has unequal side length and angle, then it 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 not more than than 180 degrees, then information technology is called 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 eight angles
The sum of the interior angles of an octagon is equal to 1080 degrees.
There are 20 diagonals in a regular octagon.

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