Polygons

Polygons come in many shapes and sizes, some with equal sides and angles, and others with different lengths and angles. They can be regular or irregular, convex or concave, and appear everywhere in daily life, from tiles and windows to signs and art.

In this guide, we will explore the types of polygons, their properties, angles, formulas for area and perimeter, common examples like triangles and quadrilaterals, and learn their names based on the number of sides.

 

Table of Contents

• What are Polygons?
• Polygon Shape
• Types of Polygons
• Angles of a Polygon
  • Interior Angles
  • Exterior Angles
• Properties of Polygons
• Area and Perimeter of Polygons
• Formulas for Common Polygons
• Triangles
• Quadrilaterals
• FAQs on Polygons

 

What are Polygons?

A polygon is a flat, closed shape made entirely of straight lines. These straight lines are called sides, and the points where two sides meet are called vertices (or corners). Think of a polygon as a shape that is like a “fence” made of straight sticks joined end to end to form a completely closed figure.

Polygons are different from other shapes because they cannot have curves. For example, a circle or an oval is not a polygon because it does not have straight sides.

 

Polygon Shape

Polygons can have different shapes and sizes. Some have equal sides and angles, while others have sides and angles of various lengths. Polygons are mainly classified according to the number of sides they have. The name of each polygon is derived from a prefix that tells how many sides it contains, which we will learn about next.

Examples of Polygon Shapes:

• Triangle

• Quadrilateral 

• Pentagon

• Hexagon 

• Heptagon

• Octagon

 

Names of Polygons

As polygons are named based on the number of sides they have. Here’s a quick table:

 

Types of Polygons

Polygons are not all the same; they come in different types depending on their sides, angles, and symmetry. Knowing the types of polygons helps us to understand shapes better and makes geometry easier to study. The main types of polygons are:

 

1. Regular Polygon

A regular polygon is a polygon where all sides are equal in length and all interior angles are equal. This makes the shape look perfectly balanced and symmetrical, which is why regular polygons are often used in architecture, art, and design.

Key Features of a Regular Polygon:

• Equal sides - every side has the same length.

• Equal angles - each angle inside the polygon is the same.

• Symmetry - you can fold or rotate it, and it still looks the same.

• Always convex - all corners point outward.

 

2. Irregular Polygon

An irregular polygon is a polygon where all sides and angles are not equal. Unlike a regular polygon, the sides may have different lengths, and the angles may be different. Because of this, irregular polygons may appear uneven and asymmetrical.

Key Features of Irregular Polygons:

• Sides are of different lengths.

• Angles are of different measures.

• Shape is asymmetrical (not balanced).

• It can be either convex or concave.

Examples of Irregular Polygons:

• A triangle with two sides equal and one side different.

• A quadrilateral with sides of different lengths and angles that are not 90°.

 

3. Convex Polygon

A convex polygon is a polygon where all interior angles are less than 180°, which means none of the sides bend inward. Every vertex “points outward,” making the polygon appear bulging or puffed out.

Key Features of Convex Polygons:

• All interior angles < 180°.

• No side points inward.

• Can be regular or irregular.

• Easy to draw and measure angles.

Examples of Convex Polygons:

• Square

• Rectangle

• Regular pentagon

• Irregular hexagon

 

4. Concave Polygon

A concave polygon is a polygon where at least one interior angle is greater than 180°, which makes a part of the polygon “cave in”. It is sometimes called a “re-entrant polygon” because of its inward fold.

Key Features of Concave Polygons:

• It has at least one interior angle > 180°.

• At least one vertex points inward.

• Usually irregular in shape.

• Looks like the polygon has a “dent” or “cut.”

Examples of Concave Polygons:

• Arrow-shaped pentagon

• Star-shaped polygons

 

Angles of a Polygon

A polygon is a closed figure made of straight line segments. It has as many vertices (corners) as it has sides. At each vertex of a polygon, an angle is formed. These angles are classified into two types:

• Interior Angles - These are the angles formed inside the polygon at its vertices.
• Exterior Angles - These are the angles formed outside the polygon when one side of the polygon is extended.

Interior Angle 

The sum of the interior angles of a polygon depends on the number of sides it has. If a polygon has n sides, then the sum of all its interior angles is given by:

Sum of interior angles = (n - 2) × 180°

Or, in radians:

Sum of interior angles = (n - 2)π radians

Example:

For a quadrilateral (n = 4):

Sum = (4 - 2) × 180°

= 2 × 180°

= 360°

This means that in any quadrilateral (square, rectangle, parallelogram, rhombus, trapezium, etc.), the sum of the four interior angles is always 360°.

 

Exterior Angle 

The interior angle and the exterior angle at the same vertex of a polygon are always supplementary. This means that:

Interior angle + Exterior angle = 180°

Another important fact is that the sum of all the exterior angles of any polygon, no matter how many sides it has, is always:

360°

Example:
For a regular pentagon (n = 5):
Each exterior angle = 360° ÷ n = 360° ÷ 5 = 72°
Then, each interior angle = 180° - 72° = 108°

 

Properties of Polygons

Polygons have several important properties that are based on their sides and angles. Understanding these makes it easier to identify and study different polygons.

• A polygon is always a closed figure, meaning all its sides join together without any gaps.

• Polygons are made up of straight line segments only. Curved lines are not allowed.

• The number of sides of a polygon is always equal to the number of corners or vertices.

• The angles inside a polygon are called interior angles. The size of these angles depends on the number of sides.

• The angles formed outside the polygon when a side is extended are called exterior angles.

• Convex polygons have all angles pointing outward.

• Concave polygons have at least one angle pointing inward, making the shape look like it has a “dent.”

• Regular polygons have all sides and angles equal.

• Irregular polygons have sides and angles of different sizes.

• A diagonal is a line connecting two non-adjacent vertices of a polygon. Polygons have one or more diagonals, depending on their number of sides.

• Many polygons, especially regular ones, have lines of symmetry, which means they can be folded or rotated and still look the same.

 

Area and Perimeter of Polygons

Polygons are flat, closed shapes, and each polygon has a specific area and perimeter depending on its sides.

• Area: The area of a polygon is the region covered by the shape on a flat surface.

• Perimeter: The perimeter is the total distance around the polygon, which you get by adding the lengths of all sides.

 

Formulas for Area and Perimeter of Common Polygons

 

Triangles

A triangle is the simplest polygon, with 3 sides and 3 vertices. Triangles are very important in geometry and are classified based on sides and angles.

Types of Triangles (Polygon)

A triangle is the simplest polygon, having 3 sides and 3 vertices. Triangles are one of the most important polygons and are classified based on sides and angles.

Triangles Based on Sides

• Equilateral Triangle
  All three sides are equal, and all three angles are equal, each measuring 60°. Also called an equiangular triangle.
  Example: A triangle with sides 5 cm, 5 cm, 5 cm.
• Isosceles Triangle
  Two sides are equal, and the angles opposite the equal sides are also equal.
  Example: A triangle with sides 6 cm, 6 cm, and 4 cm.

• Scalene Triangle
  All three sides are of different lengths, and all angles are different too.
  Example: A triangle with sides 4 cm, 5 cm, and 6 cm.

 

Triangles Based on Angles

• Acute-Angled Triangle
  All angles are less than 90°. Looks “pointed” at all corners.
  
  
• Right-Angled Triangle
  One angle is exactly 90°, forming a perfect “L” shape at the right angle.
  Example: Triangle with angles 90°, 60°, and 30°.
  
  
• Obtuse-Angled Triangle
  One angle is greater than 90°, making the triangle look “stretched” at one corner.

 

Quadrilaterals 

A quadrilateral is a polygon that has 4 sides, 4 vertices, and 4 angles. Quadrilaterals are very common in daily life and can take many different shapes, such as squares, rectangles, and trapezoids.

Types of Quadrilaterals

• Square
  All four sides are equal, and all four angles are 90°. The diagonals are equal and perpendicular, and they bisect each other.
  Example: A square-shaped chessboard tile.
  
  
• Rectangle
  Opposite sides are equal, and all angles are 90°. The diagonals are equal.
  Example: A rectangular door or window.
  
  
• Parallelogram
  Opposite sides are parallel and equal, and opposite angles are equal. The diagonals bisect each other but are not necessarily equal.
  Example: A slanted table top.
  
  
• Rhombus
  All four sides are equal, and opposite sides are parallel. The diagonals are perpendicular and bisect the angles.
  Example: A diamond-shaped kite.
  
  
• Trapezium (Trapezoid)
  Only one pair of opposite sides is parallel, and the other sides are non-parallel. Angles and diagonals vary depending on the shape.
  Example: A trapezoid-shaped bridge support or tabletop.

 

Conclusion

Polygons are an important part of geometry and are found all around us in daily life, from tiles and windows to signs and art. Understanding polygons helps us recognize shapes, calculate areas and perimeters, and study their angles and properties. Polygons can be regular or irregular, convex or concave, and each type has unique characteristics that make it useful in real life. Triangles and quadrilaterals are the simplest and most commonly used polygons, while higher-sided polygons like pentagons, hexagons, and octagons appear in designs and structures.

 

Frequently Asked Questions on Polygons

1. What is a polygon shape?

Answer: A polygon is a closed, flat shape made of straight line segments. The line segments are called sides, and the points where the sides meet are called vertices or corners. Polygons can have different numbers of sides and can be regular (all sides and angles equal) or irregular (sides and angles not equal).

 

2. How many sides does a polygon have?

Answer: A polygon can have three or more sides. There is no upper limit, but a polygon with less than three sides is not considered a polygon.

 

3. Is a circle a polygon?

Answer: No, a circle is not a polygon. A polygon is defined as a closed shape made entirely of straight sides, and it has vertices or corners where the sides meet. A circle, on the other hand, is a closed curve with a smooth edge and no straight sides or corners. Because it does not have line segments forming its boundary, it cannot be classified as a polygon.

 

4. Is a polygon 7-sided?

Answer: Yes! A 7-sided polygon is called a heptagon. It has 7 sides, 7 vertices, and the sum of its interior angles is 900°.

 

5. What is an 8-sided polygon called?

Answer: An 8-sided polygon is called an octagon. It has 8 sides, 8 vertices, and the sum of its interior angles is 1080°.

 

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