Octagon Shape Definition Properties and Formulas

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