Octagon Formula

Bookmark added to your notes.
View Notes
×

Area of Regular Octagon Formula

An octagon is a type of polygon with 8 sides; a polygon being a 2D closed plane object composed of straight-line segments with three or more sides. The term octagon is derived from the ancient greek oktágōnon, “eight angles”. A regular polygon consists of 8 sides of equal length as well as equal internal angles. On the other hand, in an irregular octagon, there are 8 different sides. Irregular polygons are not regarded as consisting of a centre and are asymmetric as well. That said, area of the octagon formula = 8 x Area of Triangle.


Octagon Area Formula

We also have another way of calculating the area of an octagonal figure using the formula for the area of a regular octagon with sides of length "a" is 2a² (1 + √2) sq. Units.


Example of an Octagon

The STOP sign used in many countries is made in the shape of an octagon.

Houses and rooms are oftentimes constructed in an octagonal shape, possibly because they somewhat appear like a circular space but are actually made up of straight wall sections which are easier to construct.

There are exemplary illustrations that are being preserved as historic buildings like the Loren Andrus Octagon House in Washington, Michigan USA. It was built in the year 1860.

Since such structures are based on a regular octagon, opposite walls are parallel and the rooms have an attractive symmetry.

Another example of an octagon is umbrellas.


(Image will be uploaded soon)


Derivation of Area of Octagon Formula

An octagon can be seen as a geometrical object with 4 rectangles, 4 isosceles triangles in the corners and a square in the centre.

The square is of area a²

The triangles have sides a, a/√2 and a/√2, by the rule of the Pythagoras theorem. Hence, each consists of an area of a²/4.

The rectangles are of area a × √2

The sum of these 9 areas comes out to be 2 a² (1 + √2).


Properties of All octagons

Basis

 

Property

Number of diagonals

20

The number of well-defined, recognizable diagonals possible from all vertices (usually ½ n(n–3)].

Number of triangles

6

The number of triangles formed by constructing the diagonals from a given vertex (usually n–2).

Sum of interior angles

1080°

Usually 180(n–2) degrees.

 


Solved Examples

Example 1:

Evaluate the area of an octagon whose length of side measures 15 cm?

Solution:

Given,

Side of an octagon = a = 15 cm

Area of an octagon

= 2 × a2 × (1 + √2)

= 2 × 152 × (1 + √2) cm2

= 2 × 225 × (1 + 1.414) cm2

= 450 × (1 + 1.414) cm2

= 1086.3 cm2   


Example 2:

We have an octagon umbrella that has a Perimeter 32 cm. Determine the area of the signboard.

Solution: Given,

Perimeter of the umbrella = 32 cm

Perimeter of an Octagon formula = 8a

32 cm = 8a

a = 32/8 = 4 cm

Area of an Octagon = Area of the umbrella

= 2 × 42 × 2.414

= 2 × 16 × 2.414

= 77.42 cm sq.


Fun Facts About Octagon Shape

  • The term ‘Octagon’ has been originated from the Greek term ‘ὀκτάγωνον’ (oktágōnon) that means eight angles, thus how the shape acquired the name -Octagon.

  • Between the 1850s and 1900s time period, many people began constructing their houses in the shape of an octagon as it is a symbol that depicts rebirth, regeneration, infinity, eternity and transition.

FAQ (Frequently Asked Questions)

Q1. What is the Difference Between a Regular Octagon and an Irregular Octagon?

Answer: These octagons are differentiated on the basis of the following i.e.:

  • Regular Octagon: A type of octagon that has all sides and angles equal. Each angle of a regular octagon measures 135° each.

  • Irregular Octagon: A type of octagon that has all sides and all angles not equivalent. Angle measurement of an irregular octagon is not similar but remembers that they all sum to 1080°.

Q2. What is the Difference Between a Convex and Concave Octagon?

Answer: Octagon is that type of polygon which has 8 sides and 8 vertices. An octagon, like any other polygon, can also be further classified either convex or concave. In a convex octagon, none of its interior angles is greater than 180°. On the hand, a concave octagon consists of one or more of its interior angles greater than 180°. Having only the sides equal is not sufficient, in that context, seeing that the octagon may be concave, with equal sides but unequal interior angles. Such octagons are known as equilateral.

Q3. What is the Bounding Box Related to Octagon?

Answer: The bounding box of a planar shape is the smallest rectangle which circumscribes the shape entirely. The dimensions of this rectangle are typically described by the height and the width of the shape. The below-given depiction defines the bounding box of a regular octagon. Both the height and the width are described as the distance between two opposite edges. Because of the symmetry, height and width are equal, and as an outcome, the bounding box is actually a square.


(Image will be uploaded soon)