how to find the area of a octagon
A polygon is a ii-dimensional (two-D) closed figure made upward of directly line segments. In geometry, the octagon is a polygon with 8 sides. If the lengths of all the sides and the measurement of all the angles are equal, the octagon is called a regular octagon. In other words, the sides of a regular octagon are congruent. Each of the interior bending and the exterior angle measure 135° and 45° respectively, in a regular octagon. There is a predefined ready of formulas for the calculation of perimeter, and area of a regular octagon which is collectively chosen equally octagon formula. For an octagon with the length of its edge as "a", the formulas are listed below.
Besides, check: Octagon
Octagon Formulas
| Formulas for Octagon | |
|---|---|
| Area of an Octagon | 2a2(1+√2) |
| Perimeter of an Octagon | 8a |
Octagon formula helps us to compute the expanse and perimeter of octagonal objects.
Derivation of Octagon Formulas:
Consider a regular octagon with each side "a"units.
Formula for Area of an Octagon:
Area of an octagon is defined as the region occupied inside the boundary of an octagon.
In order to summate the surface area of an octagon, we divide information technology into small-scale 8 isosceles triangles. Calculate the area of one of the triangles and then nosotros can multiply by 8 to find the total surface area of the polygon.
Have i of the triangles and draw a line from the noon to the midpoint of the base to form a right angle. The base of operations of the triangle is a , the side length of the polygon and OD is the height of the triangle.
Area of the octagon is given as 8 ten Area of Triangle.
2 sin²θ = ane- cos 2θ
two cos²θ = 1+ cos 2θ
\(\begin{array}{50}tan^{ii}\theta = \frac{1-cos2\theta}{ane+cos2\theta}\\ tan^{2}(\frac{45}{2})=\frac{1-cos45}{1+cos45}\\ tan^{two}(\frac{45}{2})=\frac{1-\frac{1}{\sqrt{two}}}{one+\frac{1}{\sqrt{2}}}\\ tan^{2}(\frac{45}{2})=\frac{\sqrt{2}-1}{\sqrt{2}+1}=\frac{(\sqrt{ii}-1)^{2}}{1}\\ tan(\frac{45}{2})=\sqrt{ii}-1\\ \frac{BD}{OD}=\sqrt{2}-1\\ OD=\frac{a/2}{\sqrt{2}-one}=\frac{a}{two}(1+\sqrt{two})\end{assortment} \)
Surface area of ∆ AOB =
\(\begin{assortment}{l}\frac{ane}{2}\times AB\times OD\end{array} \)
=
\(\begin{array}{l}\frac{1}{two}\times a\times \frac{a}{2}(1+\sqrt{2})\end{array} \)
=
\(\begin{assortment}{l}\frac{a^{2}}{4}(1+\sqrt{2})\cease{assortment} \)
Surface area of the octagon = 8 x Surface area of Triangle
Area of Octagon =
\(\begin{assortment}{l}8\times \frac{a^{ii}}{iv}(1+\sqrt{2})\end{assortment} \)
Expanse of an Octagon =
\(\begin{array}{l}2a^{2}(1+\sqrt{2})\cease{assortment} \)
Formula for Perimeter of an Octagon:
Perimeter of an octagon is divers every bit the length of the boundary of the octagon. So perimeter will be the sum of the length of all sides. The formula for perimeter of an octagon is given by:
Perimeter = length of 8 sides
So, the perimeter of an Octagon = 8a
Properties of a Regular Octagon:
- It has eight sides and eight angles.
- Lengths of all the sides and the measurement of all the angles are equal.
- The total number of diagonals in a regular octagon is 20.
- The sum of all interior angles is equal to 1080 degrees, where each interior bending measures 135 degrees.
- The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 45 degrees.
Solved examples Using Octagon Formula:
Question ane: Calculate the area and perimeter of a regular octagon whose side is 2.3 cm.
Solution: Given, side of the octagon = 2.iii cm
Area of an Octagon =
\(\brainstorm{array}{fifty}2a^{ii}(ane+\sqrt{two})\end{array} \)
Surface area of an Octagon =
\(\begin{array}{l}ii\times two.3^{two}(1+\sqrt{2})=25.54\;cm^{2}\cease{array} \)
Perimeter of the octagon= 8a= 8 × 2.3 = 18.iv cm
Question 2:Perimeter of an octagonal cease signboard is 32 cm. Notice the area of the signboard.
Solution: Given,
Perimeter of the finish sign lath = 32 cm
Perimeter of an Octagon = 8a
32 cm = 8a
a = 32/eight = 4 cm
Area of an Octagon =
\(\begin{array}{l}2a^{2}(1+\sqrt{ii})\cease{assortment} \)
Expanse of the terminate sign board =
\(\begin{array}{l}2\times 4^{ii}(1+\sqrt{2})=77.248\;cm^{2}\stop{assortment} \)
To solve more problems on the topic, download BYJU'S – the Learning App.
Source: https://byjus.com/octagon-formula/
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