Octagon is an eight-sided two-dimensional geometrical figure derived from Greek words "octa," which meant "eight," and "gonon" which meant "angles. There are 20 diagonals in an octagon. An octagon consists of 8 interior angles and 8 exterior angles. The sum of an octagon's interior angles is 1080°, and the sum of the exterior angles of an octagon is 360°. Octagons are classified into various types based upon their sides and angles. Let's learn more about the octagon shape in this article.
|1.||What is an Octagon?|
|2.||Shape of Octagon|
|3.||Types of Octagons|
|4.||Properties of an Octagon|
|5.||Angles of an Octagon|
|6.||Perimeter of an Octagon|
|7.||Solved Examples on Octagon|
|8.||Practice Questions on Octagon|
|9.||FAQs on Octagon|
What is an Octagon?
An octagon can be defined as a polygon having eight sides joined together with one another, eight interior angles, and eight vertices. When all the sides and angles of an octagon are equal in measurement, it is called a regular octagon, while when it has at least two sides or angles in different measurements, it is known as an irregular octagon. Every polygon is either convex or concave. Convex octagons bulge outwards, whereas concave octagons have indentations (a deep recess). Convex octagons are those in which all the angles point outwards. A regular octagon is an example of a convex octagon. The octagon in which at least one of its angles point inwards is a concave-shaped octagon.
Shape of Octagon
An octagon can be considered as a circle attempting to become a square or a square attempting to become a circle. Octagon is a geometrical shape in a 2D plane. Octagons have straight sides that connect; they may be curved but connected. An octagon has 8 straight sides. The shape must also be closed. One of the most popular examples of the octagon is the STOP signboard.
Types of Octagons
Depending upon the sides and angles in an octagon, it is classified into the following categories:
- Regular and Irregular Octagon
- Concave and Convex Octagon
The octagon having eight congruent sides and angles is known as a regular octagon. Every regular octagon has the same angle measures.
- All sides are equal in length, and all the angles are equal in measure.
- The interior angles add up to 1080°(135x8) and the exterior angles add up to 360°.
- The interior angle at each vertex of a regular octagon is 135°. The central angle is 45°.
An octagon that does not have any congruent sides or congruent angles is an irregular octagon. An irregular Octagon has eight unequal sides and eight unequal angles.
- It is an octagon with unequal sides and angles.
- All the angles are different in measurement, but the sum is always 1080º.
The octagon, which has all its angles pointing outside and no angle pointing inwards, is convex. Each angle of a convex octagon is less than 180°.
- Convex octagons bulge outward.
- None of their interior angles is greater than 180°
The octagon in which one of the angles pointing inward is a concave-shaped octagon.
- Concave octagons have indentations (a deep recess).
- The interior angles are greater than 180°
Properties of an Octagon
Here are a few properties of a regular octagon related to its angles, sides, and diagonals.
- It has eight equal sides and eight equal angles.
- All its interior angles sum upto 1080°
- The measure of its central angles equals 45°
- The number of triangles in a regular octagon that is formed by drawing the diagonals from a given vertex to the remaining vertices is 6
- It has 20 diagonals.
Angles of an Octagon
We have already discussed the angles of a regular octagon in the section: Properties of octagons. In a regular octagon, the vertex's interior angle is 135°, and the central angle is 45°. The sum of an octagon's interior angles is 1080°, and the sum of the exterior angles of an octagon is 360°.
- Sum of Interior angles = 135° × 8 sides = 1080°
- Sum of Exterior angles = 45° × 8 sides = 360°
Perimeter of an Octagon
To calculate the perimeter, the length of all sides should be known. For a regular octagon, the length of all sides is the same. As per the perimeter of the octagon formula, multiply 8 with the side length to know the perimeter value.
The perimeter of a regular octagon = Sum of all its sides = 8a (Where "a" is the length of one side of the octagon)
- An octagon has eight sides.
- The sum of all the interior angles in an octagon is always 1080º.
- The sum of all the exterior angles in an octagon is always 360º.
- A regular octagon has 20 diagonals.
- Regular octagons are always convex octagons, while irregular octagons can either be concave or convex.
Thinking Out of the Box!
- Explore a few famous octagonal buildings and structures around the world.
- What is the sum of the number of sides, vertices, and angles in an irregular octagon?
a) 10 b) 8 c) 30 d) 40
- What is the number of triangles that can be drawn in a regular octagon by connecting all the vertices?
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Solved Examples on Octagon
Example 1: Lucas is curious to know the number of triangles formed by joining the vertices of an octagon. Let's help him!
In a regular octagon, by joining one vertex to the remaining vertices, 6 triangles can be formed.
Similarly, if you are considering all the vertices independently, you will have a total of 632 triangles.
Therefore, 632 triangles in all.
Example 2: Noah was solving a Math sum which required him to calculate the length of each side of a regular octagon while the perimeter of the octagon was given as 160 units. Let's help him out!
According to the regular octagon definition, all of its sides are of equal length. The perimeter of an octagon = 8 × (side). Here, the perimeter is given as 160 units. Thus, the length of each side = 160 ÷ 8 = 20 units. Therefore, the length of each side of an octagon whose perimeter is 160 units is 20 units.
Example 3: Alex is planning to construct an octagonal-shaped garden in front of his house. As per his plan, each side of the octagonal garden will measure 10 units. He needs to buy a fence wire to place around the boundary of the garden. Find the length of the fence wire required.
We know that the garden is octagonal shaped. Length of the side of the octagonal garden = 10 units. To find the length of the fence wire required, we need to find the garden's perimeter. We already know that the perimeter of an octagon = 8 × side. Hence, the perimeter of the garden = (10 × 8) units = 80 units
Therefore, 80 units of fence wire is needed.
FAQs on Octagon
How many Sides are there in an Octagon?
There are eight sides in an octagon. When all these eight sides are equal in length, it is known as a regular octagon, whereas when even at least one of the sides is different in measurement, it is known as an irregular octagon.
Is an Octagon a Quadrilateral?
No, an octagon is not a quadrilateral. A quadrilateral is a closed shape with four vertices and four sides and an octagon has 8 sides and 8 vertices.
What is the Angle Measure of the Angles that form the Corners of an Octagon?
The interior angle at each vertex of a regular octagon is 135°. The central angle is 45°.
Is any 8-Sided Shape an Octagon?
If the shape is closed, made up of straight lines, and has eight sides, we call it an octagon.
Do Octagons have Equal Sides?
All octagons need not have equal sides. Octagons that have equal sides are known as regular octagons. On the other hand, irregular octagons have different side lengths.
Where do the Longest Diagonal of an Octagon Meet?
In total, an octagon has 20 diagonals; the longest ones are on its axes of symmetry, and they meet at the central point, which is also the origin of symmetry.
How many Parallel Lines does an Octagon have?
A regular octagon has 4 pairs of parallel sides. There are a total of 8 sides in an octagon, and those eight sides are parallel to their respective opposite side in the case of a regular octagon.