Polygons Explained | Types, Properties & Key Formulas

Polygons are an essential part of geometry and appear frequently in both theoretical and practical applications. Whether you’re a student learning geometry, preparing for competitive exams, or simply curious about shapes, understanding polygons is crucial. This article explores the fundamental properties of polygons, with a special focus on regular polygons like hexagons and octagons.

What is a Polygon?

A polygon is a plane figure that is formed by a finite chain of straight-line segments, creating a closed circuit. The individual line segments are called the edges or sides, and the points where two edges meet are known as the vertices (singular: vertex).

In simple terms, a polygon is a closed shape made by connecting several straight lines end to end.

Regular Polygons: Key Features and Formulas

A regular polygon is a special type of polygon where:

All sides are of equal length.
All interior angles are equal.

The vertices of a regular polygon lie on a circle, and this circle’s center is also the center of the polygon. Understanding the properties of regular polygons is essential for solving many geometry problems.

Formulas for Regular Polygons

Central Angle:
The central angle (angle between two sides at the center) of a regular polygon with n sides is given by:
$Θ=360∘n\Theta = \frac{360^\circ}{n}$ Where n is the number of sides of the polygon.
Interior Angle:
The interior angle of a regular polygon is calculated using the formula:
$angle=(n−2)×180∘n\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$ This formula gives the measure of each interior angle of a regular polygon with n sides.
Sum of Interior Angles:
The sum of all interior angles of an n-sided regular polygon is:
$angles=180∘×(n−2)\text{Sum of interior angles} = 180^\circ \times (n-2)$ This formula is helpful for finding the total angle measure when working with polygons.

Exploring Regular Hexagons

A regular hexagon is a polygon with six equal sides. It is a special type of regular polygon because of its symmetry and unique properties.

Key Properties of a Regular Hexagon:

Equal Sides: All sides of a regular hexagon have the same length.
Equilateral Triangles: When you join each vertex to the center of the hexagon, you create 6 equilateral triangles.
Area of Regular Hexagon:
The area of a regular hexagon with side length a is given by: $Area=6×34×a2\text{Area} = 6 \times \frac{\sqrt{3}}{4} \times a^2$

This formula shows that the area of a hexagon is directly related to the square of its side length, making it a powerful tool in geometry.

Exploring Regular Octagons

A regular octagon is a polygon with eight equal sides and eight equal interior angles. Octagons are often seen in real life, such as in stop signs and architectural designs.

Key Properties of a Regular Octagon:

Sum of Interior Angles:
The sum of all interior angles of a regular octagon is:
$angles=180∘×(8−2)=1080∘\text{Sum of angles} = 180^\circ \times (8 – 2) = 1080^\circ$ Each interior angle is:

$angle=1080∘8=135∘\text{Interior angle} = \frac{1080^\circ}{8} = 135^\circ$
Isosceles Triangles:
When you join each vertex of a regular octagon to the center, you create 8 isosceles triangles.

Conclusion:

Understanding polygons, particularly regular polygons like hexagons and octagons, is crucial in both academic and real-world applications. The ability to calculate angles, areas, and other properties of these shapes will help you excel in geometry problems. Whether you’re studying for exams or exploring geometry concepts for fun, these fundamental properties and formulas will serve as a solid foundation.