How to Find the Area & Perimeter of a Regular Octagon?
The concept of octagon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are preparing for school exams or competitive Olympiads, understanding octagons helps you solve questions on polygons, area and perimeter, and geometry word problems. On this page, you will learn the definition, properties, formulas, and practical examples of octagons in a student-friendly, easy-to-read manner.
What Is Octagon?
An octagon is defined as a polygon with eight sides and eight angles. Every octagon has eight straight sides that join to form a closed figure. You’ll find this concept applied in areas such as geometry, measurement, and real-life pattern recognition. Regular octagons have sides and angles that are all equal, and irregular octagons have different side or angle lengths.
Octagon Properties Table
| Property | Regular Octagon | Irregular Octagon |
|---|---|---|
| Number of sides | 8 | 8 |
| Number of angles | 8 (each 135°) | 8 (angles may vary) |
| Length of sides | All equal | Not all equal |
| Sum of interior angles | 1080° | 1080° |
| Number of diagonals | 20 | 20 |
| Symmetry | 8 lines | Typically none |
Key Formula for Octagon
Here are the standard formulas every student should remember for octagons:
| Formula | Expression |
|---|---|
| Area of a regular octagon | \( 2(1 + \sqrt{2})\,a^2 \) where a = side length |
| Perimeter of a regular octagon | \( 8a \) |
| Sum of interior angles | \( (8-2)\times180^\circ = 1080^\circ \) |
| Each interior angle (regular) | \( 135^\circ \) |
| Number of diagonals | \( \frac{8(8-3)}{2} = 20 \) |
Cross-Disciplinary Usage
Octagon is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NTSE, or NEET will see its relevance in questions involving polygons, geometric design, and reasoning puzzles. Octagons also appear in architecture and road signs, like the famous stop sign.
Types of Octagons: Regular, Irregular, Convex, Concave
Regular octagon: All sides and angles are the same. Example - a stop sign.
Irregular octagon: Sides or angles are not all equal.
Convex octagon: All angles less than 180°, no sides cave inward.
Concave octagon: At least one angle more than 180°; figure caves in at one or more points.
Step-by-Step Illustration: Find Area & Perimeter
Let’s solve a sample problem on a regular octagon whose side is 7 cm.
1. Given: Side of octagon, \( a = 7\,\text{cm} \)2. Find perimeter:
Perimeter = 8 × 7 = 56 cm
3. Find area:
Area = \( 2(1 + \sqrt{2}) \times 7^2 \)
First, \( 7^2 = 49 \), \( \sqrt{2} \approx 1.414 \)
So \( 1 + 1.414 = 2.414 \), \( 2 \times 2.414 = 4.828 \)
Final area = \( 4.828 \times 49 = \) 236.572 cm2
Speed Trick or Vedic Shortcut
To quickly remember the area formula for a regular octagon, use the shortcut: multiply the square of the side by approximately 4.828 to get the area. For calculations in competitive exams or during self-practice, estimating \( \sqrt{2} \) as 1.414 helps save time.
Try These Yourself
- Draw and label a regular octagon, marking all sides and angles.
- Find the area of a regular octagon with side 5 cm.
- How many diagonals does an octagon have?
- Compare a regular octagon and hexagon in terms of sides and angle sum.
Frequent Errors and Misunderstandings
- Mixing up a regular octagon with just any eight-sided figure.
- Using the area formula for regular octagon on irregular ones—remember, it only works when all sides/angles are equal.
- Forgetting to multiply final area by 2(1 + √2), using only 8 × a or a simple square area.
Relation to Other Concepts
The idea of octagon connects closely with topics such as polygons and area of polygon. Mastering octagons makes it easier to tackle MCQs and word problems on polygons, shape properties, and even symmetry questions. You can also check symmetry or diagonals for more related learning.
Classroom Tip
A quick way to remember octagon facts is the “OCT” in octagon stands for “eight”: 8 sides, 8 angles, and the area formula starts with ‘8’ (since 8/2 = 4 in the apothem-based formula). Vedantu’s teachers show how octagons look in real life (stop signs, tiles, shapes) to help students visualize and recall the concept.
Wrapping It All Up
We explored octagon—from definition, formula, examples, frequent exam mistakes, and its links to polygons, symmetry, and real-world objects. Keep practicing with Vedantu’s revision questions and video classes to master solving octagon problems easily and score confidently in your maths exams!
Explore More: Polygons | Area of Polygon | Regular Polygon | Perimeter of a Polygon | Symmetry
FAQs on Octagon – Definition, Properties & Examples
1. What is an octagon in maths?
An octagon is a polygon with eight sides and eight angles. When all sides and angles are equal, it's called a regular octagon; otherwise, it's an irregular octagon.
2. How many sides and angles does an octagon have?
An octagon always has eight sides and eight angles.
3. What is the area formula for a regular octagon?
The area of a regular octagon with side length 'a' is calculated using the formula: Area = 2(1 + √2)a². This formula is specifically for regular octagons where all sides are equal.
4. How do you find the perimeter of an octagon?
The perimeter of any octagon is the sum of the lengths of its eight sides. For a regular octagon, the perimeter is simply 8 times the length of one side.
5. What is the sum of all interior angles of an octagon?
The sum of the interior angles of any octagon is 1080°. This applies to both regular and irregular octagons.
6. Where do we see octagons in real life?
Octagons appear in various real-world objects, including stop signs, some building designs, and certain types of tiles.
7. What is the difference between a regular and irregular octagon?
A regular octagon has all eight sides and angles equal in measure. An irregular octagon has unequal sides and/or angles.
8. How many diagonals does an octagon have?
An octagon has 20 diagonals. This can be calculated using the formula n(n-3)/2, where 'n' is the number of sides (8 in this case).
9. What is the measure of each interior angle in a regular octagon?
Each interior angle of a regular octagon measures 135°.
10. What is the measure of each exterior angle in a regular octagon?
Each exterior angle of a regular octagon measures 45°.
11. Can an octagon be concave?
Yes, an octagon can be concave. A concave octagon has at least one interior angle greater than 180°.
12. How is the apothem of a regular octagon used in area calculations?
The apothem (the distance from the center to the midpoint of a side) of a regular octagon can be used in an alternative area formula: Area = (1/2) * perimeter * apothem.
