How to Calculate the Area and Perimeter of an Octagon
Any 2-dimensional shape with more than 2 line segments is called a polygon. Some known polygon examples include square, triangle, rectangle, pentagon, etc. Likewise, polygons with 8 sides in a 2-dimensional plane are known as an octagon. These shapes can again be divided into categories depending on their side and angle measurements. In this section, students will learn about the different types of octagons and how that affects their properties and calculations, including the area of an octagon.
Read on to get acquainted with the basics before heading on with the formulae.
Types of Octagons
Octagons can be divided into 2 types, based on their side lengths.
Regular Octagon:
An octagon with equal-length sides and all the same value angles is called a regular octagon.
Irregular Octagon:
If octagon sides and angles are unequal, it is called an irregular octagon.
Properties of a Regular Octagon
Following are the features of a regular octagon.
It comprises 8 sides and 8 angles.
The measurement of each side is equal, as is the case with its angles.
A regular octagon comprises a total of 20 diagonals.
Each interior angle measures 135°, summing up to 1080° degrees.
Each of a regular octagon’s exterior angles measures 45°, giving a total sum of 360°.
Based on the above characteristics, students can derive many other properties of an octagon, which they will need to solve the varied numerical that they will encounter under this topic. Read on to know how to derive some of these like area, perimeter, and diagonal formulas for an octagon.
Area of a Regular Octagon
The most widely used formula to calculate the area of a regular octagon is given as:
A = 2a2 (1 + √2), where a represents the given octagon’s each side length.
To derive this equation, consider the given pentagon. Drawing all its diagonals has divided it into 8 isosceles triangles with the center as their common apex.
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Calculating the area of one of these similar triangles, and multiplying it by 8 will give you the given octagon’s area. To calculate a triangle’s area, draw a perpendicular line OP joining the apex to base AB’s bisectional point, as shown in the figure below. This is an apothem of an octagon, which can be calculated by dividing any of the longest diagonals by 2.
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Here, AB = side of given octagon = a
Therefore AP = PB = a/2
Now, angle OPB = angle OPA = 90°
And angle OAP = 135/2°, and angle AOP = θ = 45/2°, by construction.
To determine OD’s length, solve trigonometric expressions for θ.
2 sin² θ = 1 - cos 2θ – (i)
2 cos² θ = 1 + cos 2θ – (ii)
Divide (i) by (ii),
tan2 θ = (1 - cos 2θ) / (1 + cos 2θ) – (iii)
Put the value of θ in equation (iii),
tan2 (45/2) = (1 - cos 45) / (1 + cos 45)
Substitute the value of cos 45 as 1/√2,
tan2 (45/2) = (1 – 1/√2) / (1 + 1/√2) = (√2 – 1) / (√2 + 1) = (√2 – 1)2/1
tan (45/2) = √2 - 1
Since angle AOP = 45/2°, tan (45/2) = AP/OP
i.e., AP/OP= √2 - 1
or, OP = AP / (√2 – 1) = (a/2) / (√2 – 1) = (a/2) (1+√2)
Therefore, area of ∆ OAB = (1/2) x AB x OP = (1/2) x a x (a/2) (1+√2) = (a2/4) (1+√2)
As discussed earlier, area of regular octagon= 8 x area of ∆ OAB = 8 x (a2/4) (1+√2)
Hence, proved the area of an octagon = 2a2 (1 + √2).
Area of an Irregular Octagon
The above formulae to calculate the area of an octagon were for the condition that every side and angle are equal. For irregular shapes like the one given below, the previous formulae are obsolete.
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In this scenario, to calculate the area of an octagon formula, you need to visualize a given octagon divided into different polygons, whose necessary measurements are known or can be derived from given information. Now, calculate each of these polygons’ area with their known area formulae. Lastly, add calculated areas of these polygons to find the area of the octagon.
The formula to find the area of some common polygons, besides triangle are given below.
Area of square = s2, where s represents each side’s length.
Area of rectangle = l x b, where l stands for length and b represents width.
Area of parallelogram = b x h, where b is its base length, and h is its height.
The perimeter of an Octagon
The total distance covered by an octagon’s periphery is called its perimeter. In other words, it is the length of its boundary. Therefore calculating the perimeter of the octagon formula is nothing but a sum of the measurement of all its sides. Therefore, it can be represented as:
The perimeter of octagon = 8a, where a is the given polygon’s each side length.
Length of a Diagonal of an Octagon
The diagonal of an octagon is a line drawn to join any pair of its opposite vertices. A regular octagon comprises 4 equal diagonals, each of which divides it into two similar pentagons. The formula to calculate their length is given as:
L = a √( 4 + 2√2), where a is the length of each side of the octagon.
The above formula can be derived as follows.
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In ∆ AED in given diagram, AE2 + DE2 = AD2, (Pythagoras’ theorem for right-angled triangle)
i.e., b2 + b2 = a2
⇒ b = a / √2 – (iv)
Now, AB = 2b + a = a√2 {substituting b’s value form (iv)}, and BC = a.
In ∆ABC, AC2 = AB2 +BC2
Therefore, length of diagonal AC = a √(4 + 2√2).
Now, you can implement diagonal length calculated by this formula to find the area of an octagon.
The Area of the octagon for class 8 is an important concept to help students develop a clear foundation of geometry for efficient numerical solving as well as higher studies. A thorough study of prescribed textbooks undoubtedly comes first for exam preparation. For a more detailed understanding of this topic along with illustrations and exercises on different types of problem sums, refer to online tutoring sites like Vedantu. Download the app to avail of live expert guidance.
Properties of an Octagon
The area of an octagon is studied much earlier in classes, however, it is mainly introduced in the eighth grade in chapter 3 called understanding quadrilaterals. This chapter is mainly concerned with the various types of polygons and their classification. For example, some of the polygons are triangle, quadrilateral, Pentagon, hexagon, heptagon, octagon, nonagon, decagon. This chapter also explains in-depth diagonals, convex and concave polygons, regular and irregular polygons, angles some properties, some of the measures of the exterior angles of a polygon. After studying these figures in-depth this chapter further explains the kinds of quadrilaterals like trapezium, kite, parallelogram, elements of a parallelogram, angles of a parallelogram, diagonals of a parallelogram, this chapter goes on to further explain some special parallelogram such as rhombus, rectangle, a square.
An octagon is a type of polygon, a polygon can be defined as a figure having a two-dimensional closed figure which is made of straight line segments having three or more sides. An octagon is a figure having eight sides. The word octagon originated from greek octagonal, which means “eight angles”. There are two types of octagons, regular and irregular octagons. A regular octagon mainly consists of eight sides having an equal length and equal internal angles that is 135° and internal angles being 45°. And a regular octagon has eight different sides, these types of figures are not symmetric and they do not have a center. There are many examples of an octagon that can be seen in our day-to-day life for example the stop sign used in many places is usually in the shape of an octagon.
Get an Edge with Vedantu
The article mainly discusses in detail the types of octagons that are regular and irregular octagon, properties of a regular octagon, area of a regular octagon, area of an irregular octagon, perimeter of an octagon, length of a diagonal of an octagon. These are the basic concepts that are all mandatory for students to understand and get a good hold over them as it is only after getting The basics cleared that a student can excel further in studies. The objective of this article is to make students efficient in solving the area and perimeter of an octagon.
Finding the area for a long time can be a difficult task for some students and in order to make the learning process easier Vedantu’s expert mathematics teachers who have years of experience have done extensive research on the topic area of the octagon and they have explained this concept in an extremely simplified manner so that students who find it difficult can also get a good grasp over the calculations regarding an octagon.
FAQs on Area of an Octagon: Complete Guide for Students
1. What is the formula for the area of a regular octagon?
The area of a regular octagon with side length $a$ can be calculated by the formula $A = 2(1 + \sqrt{2})a^2$. This formula uses the side length to find the total space inside the octagon.
2. How do you find the area of an octagon with a given side length?
To calculate the area of an octagon, substitute the side length $a$ into the formula $A = 2(1 + \sqrt{2})a^2$. Multiply the value inside the parentheses by 2, then by $a^2$ to get the area in square units.
3. What are the steps to calculate the area of a regular octagon?
Follow these steps to find the area of a regular octagon:
- Find the side length $a$.
- Calculate $a^2$.
- Multiply by $(1 + \sqrt{2})$.
- Multiply the result by 2.
4. Can you find the area of a regular octagon using its apothem?
Yes, the area of an octagon can also be calculated using its apothem $a$ and perimeter $P$ by $A = \frac{1}{2}Pa$. For a regular octagon, this method is accurate because all sides and angles are equal.
5. How does the area of an octagon compare to a square with the same side length?
A square's area is $a^2$, while a regular octagon's area is $2(1 + \sqrt{2})a^2$, which is about 2.828 times larger. The octagon includes more space because of the extra sides.
6. What units are used for the area of an octagon?
The area of an octagon is always measured in square units, such as square centimeters (cm²), square meters (m²), or square inches (in²), depending on the side length's unit.
7. How do you find the area of an irregular octagon?
For an irregular octagon, divide it into simpler shapes like triangles or rectangles, find each area separately, then add them up. This method helps when sides are not equal or angles differ.
8. Why is it important to know the area of an octagon?
Knowing the area of an octagon is useful in design, construction, and land measurement. It helps calculate how much material is needed or the usable space inside an eight-sided structure.
9. What is the relationship between side length and area for a regular octagon?
For a regular octagon, the area increases as the square of the side length. If you double the side length, the area becomes four times larger, as shown by $A = 2(1 + \sqrt{2})a^2$.
10. Can you calculate the area of an octagon with only the circumradius?
Yes, the area of a regular octagon with circumradius $R$ is $A = 2R^2\sqrt{2}$. This formula uses the distance from the center to a vertex, making calculations possible when side length is unavailable.
11. How does symmetry help in finding the area of a regular octagon?
A regular octagon has equal sides and angles, letting us use special formulas for area. Its symmetry means you can divide it into equal triangles, simplifying calculations and confirming results for accuracy.
12. What is the area if the side of a regular octagon is 5 cm?
Using the formula for area of a regular octagon, $A = 2(1 + \sqrt{2})a^2$, plug in $a = 5$ cm. Calculate: $A = 2(1 + 1.4142)\times25 \approx 2(2.4142)\times25 \approx 120.71$ cm².
