Introduction to Polygons
Look around your classroom. The whiteboard is a rectangle. The clock might be a circle. But what about a stop sign? It has 8 sides. What about a honeycomb cell? It has 6 sides. These many-sided shapes are called polygons.
The word polygon comes from the Greek words poly (meaning "many") and gon (meaning "angles" or "corners"). So a polygon is a shape with many angles and many sides.
You already know some polygons: triangles (3 sides), rectangles and squares (4 sides). But there are polygons with 5, 6, 7, 8, 10, and even more sides. Each has its own name and properties.
In this chapter, you will learn what a polygon is, how to name polygons based on the number of sides, the difference between regular and irregular polygons, and how to count sides, vertices, and diagonals.
What is Introduction to Polygons?
Definition: A polygon is a closed figure made up of straight line segments that do not cross each other.
Conditions for a shape to be a polygon:
- It must be a closed figure (the lines must connect end to end to form a complete boundary).
- It must be made of straight lines only (no curves).
- The sides must not cross each other.
- It must have at least 3 sides (a triangle is the simplest polygon).
Parts of a Polygon:
| Part | Meaning | Example |
|---|---|---|
| Side | One of the straight line segments that form the polygon | A triangle has 3 sides |
| Vertex | A corner point where two sides meet (plural: vertices) | A triangle has 3 vertices |
| Angle | The space between two sides at a vertex | A triangle has 3 angles |
| Diagonal | A line segment connecting two non-adjacent vertices | A rectangle has 2 diagonals |
What is NOT a polygon:
- A circle (curved, not made of straight lines).
- An open figure (sides do not form a closed shape).
- A figure with crossing sides (like a star shape drawn with crossing lines).
Introduction to Polygons Formula
Counting Parts of a Polygon:
In any polygon with n sides:
Number of sides = Number of vertices = Number of angles = n
Number of diagonals = n(n - 3) / 2
Where:
- n = number of sides
Diagonal Count Examples:
| Polygon | Sides (n) | Diagonals = n(n-3)/2 |
|---|---|---|
| Triangle | 3 | 3(3-3)/2 = 0 |
| Quadrilateral | 4 | 4(4-3)/2 = 2 |
| Pentagon | 5 | 5(5-3)/2 = 5 |
| Hexagon | 6 | 6(6-3)/2 = 9 |
| Octagon | 8 | 8(8-3)/2 = 20 |
| Decagon | 10 | 10(10-3)/2 = 35 |
Sum of Interior Angles:
Sum of interior angles = (n - 2) x 180 degrees
Types and Properties
Polygons Named by Number of Sides:
| Sides | Name | Real-Life Example |
|---|---|---|
| 3 | Triangle | Yield sign, slice of pizza |
| 4 | Quadrilateral | Book, door, TV screen |
| 5 | Pentagon | The Pentagon building (USA), some football patterns |
| 6 | Hexagon | Honeycomb cell, nuts and bolts |
| 7 | Heptagon | Some coins (UK 50p coin shape) |
| 8 | Octagon | Stop sign |
| 9 | Nonagon | Some decorative tiles |
| 10 | Decagon | Some coins, decorative patterns |
Regular vs Irregular Polygons:
- A regular polygon has all sides equal in length AND all angles equal in measure. Example: a square (all sides equal, all angles 90 degrees), an equilateral triangle.
- An irregular polygon has sides of different lengths OR angles of different measures (or both). Example: a rectangle is an irregular polygon because not all sides are equal (though all angles are equal).
Convex vs Concave Polygons:
- A convex polygon has all interior angles less than 180 degrees. All diagonals lie inside the polygon. This is the "normal" polygon shape.
- A concave polygon has at least one interior angle greater than 180 degrees. It looks like it has a "dent" or "cave" pushed inward. At least one diagonal lies outside the polygon.
Types of Quadrilaterals (4-sided polygons):
- Square, Rectangle, Parallelogram, Rhombus, Trapezium, Kite
Solved Examples
Example 1: Identify the Polygon
Problem: A shape has 5 sides. What polygon is it?
Solution:
- A polygon with 5 sides is called a pentagon.
- "Penta" means five.
Answer: It is a pentagon.
Example 2: Count Vertices
Problem: How many vertices does a hexagon have?
Solution:
- A hexagon has 6 sides.
- Number of vertices = Number of sides = 6.
Answer: A hexagon has 6 vertices.
Example 3: Count Diagonals of a Quadrilateral
Problem: How many diagonals does a quadrilateral have?
Solution:
Given:
- n = 4 (quadrilateral has 4 sides)
Using the formula:
- Diagonals = n(n - 3) / 2
- = 4(4 - 3) / 2
- = 4 x 1 / 2
- = 4 / 2
- = 2
Answer: A quadrilateral has 2 diagonals.
Example 4: Count Diagonals of a Pentagon
Problem: How many diagonals does a pentagon have?
Solution:
Given:
- n = 5
Using the formula:
- Diagonals = 5(5 - 3) / 2
- = 5 x 2 / 2
- = 10 / 2
- = 5
Answer: A pentagon has 5 diagonals.
Example 5: Sum of Angles of a Triangle
Problem: What is the sum of interior angles of a triangle?
Solution:
Given:
- n = 3 (triangle)
Using the formula:
- Sum = (n - 2) x 180
- = (3 - 2) x 180
- = 1 x 180
- = 180 degrees
Answer: The sum of interior angles of a triangle is 180 degrees.
Example 6: Sum of Angles of a Hexagon
Problem: What is the sum of interior angles of a hexagon?
Solution:
Given:
- n = 6 (hexagon)
Using the formula:
- Sum = (n - 2) x 180
- = (6 - 2) x 180
- = 4 x 180
- = 720 degrees
Answer: The sum of interior angles of a hexagon is 720 degrees.
Example 7: Regular or Irregular?
Problem: A quadrilateral has all sides of length 5 cm and all angles of 90 degrees. Is it a regular polygon?
Solution:
Check:
- All sides are equal (5 cm each). Yes.
- All angles are equal (90 degrees each). Yes.
Since both conditions are met:
- It is a regular polygon.
- A regular quadrilateral with all angles 90 degrees is a square.
Answer: Yes, it is a regular polygon (a square).
Example 8: Is a Rectangle Regular?
Problem: Is a rectangle a regular polygon?
Solution:
Check:
- All angles are equal (90 degrees each). Yes.
- All sides are equal? A rectangle has opposite sides equal, but not all four sides are equal (unless it is a square). No.
Since not all sides are equal:
- A rectangle is an irregular polygon.
Answer: No, a rectangle is NOT a regular polygon (unless it is a square).
Example 9: Naming a Polygon from Description
Problem: A polygon has 8 equal sides and 8 equal angles. Name it and give a real-life example.
Solution:
- 8 sides = octagon.
- All sides and all angles are equal = regular octagon.
- Real-life example: Stop sign on roads.
Answer: It is a regular octagon. Example: a stop sign.
Example 10: Finding Sides from Diagonals
Problem: A polygon has 9 diagonals. How many sides does it have?
Solution:
Given:
- Diagonals = 9
- n(n - 3) / 2 = 9
Solving:
- n(n - 3) = 18
- Try n = 6: 6 x 3 = 18. Yes!
Answer: The polygon has 6 sides. It is a hexagon.
Real-World Applications
Polygons are found everywhere in the real world:
Architecture: Buildings use rectangular, triangular, and hexagonal shapes. The Pentagon building in the USA is a famous pentagon-shaped building.
Nature: Honeycomb cells are perfect regular hexagons. Snowflakes have 6-sided symmetry. Many crystals are shaped like polygons.
Road Signs: Stop signs are octagons. Yield signs are triangles. Speed limit signs are rectangles. The shape helps drivers recognise the sign even from far away.
Sports: A football (soccer ball) is made of pentagons and hexagons stitched together.
Floor Tiles: Tiles are often squares, rectangles, or hexagons because these shapes can cover a floor without leaving gaps. This is called tessellation.
Art and Design: Patterns in rangoli, kolam, and Islamic art use polygons extensively. Logo designers often use regular polygons for their balanced, pleasing shapes.
Key Points to Remember
- A polygon is a closed figure made of straight line segments that do not cross each other.
- Polygons are named by the number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
- Number of sides = Number of vertices = Number of angles.
- Number of diagonals = n(n - 3) / 2.
- Sum of interior angles = (n - 2) x 180 degrees.
- A regular polygon has all sides equal AND all angles equal.
- An irregular polygon has unequal sides or unequal angles (or both).
- A convex polygon has all interior angles less than 180 degrees.
- A concave polygon has at least one interior angle greater than 180 degrees.
- The simplest polygon is a triangle (3 sides). A triangle has 0 diagonals.
Practice Problems
- Name the polygon with 7 sides.
- How many diagonals does an octagon have?
- What is the sum of interior angles of a pentagon?
- Is an equilateral triangle a regular polygon? Why or why not?
- A polygon has 20 diagonals. How many sides does it have?
- Name three objects around you that are shaped like polygons. State how many sides each has.
- Can a polygon have 2 sides? Explain why or why not.
- What is the difference between a regular and an irregular polygon? Give an example of each.
Frequently Asked Questions
Q1. What is a polygon?
A polygon is a closed figure made of straight line segments. The sides do not cross each other, and the figure must have at least 3 sides. Examples include triangles, squares, pentagons, and hexagons.
Q2. Is a circle a polygon?
No. A circle is not a polygon because it is made of a curved line, not straight line segments. Polygons must have only straight sides.
Q3. What is the smallest polygon?
A triangle (3 sides) is the smallest polygon. You cannot make a closed figure with fewer than 3 straight lines.
Q4. What is the difference between regular and irregular polygons?
A regular polygon has all sides equal and all angles equal (like an equilateral triangle or a square). An irregular polygon has sides of different lengths or angles of different measures (like a rectangle or a scalene triangle).
Q5. How many diagonals does a triangle have?
A triangle has 0 diagonals. Each vertex is already connected to both other vertices by a side, so there are no non-adjacent vertices to connect.
Q6. What is a diagonal?
A diagonal is a line segment that connects two non-adjacent vertices (corners that are not next to each other) of a polygon. For example, in a rectangle, the line from one corner to the opposite corner is a diagonal.
Q7. Why are stop signs shaped like octagons?
Stop signs are octagons (8 sides) so drivers can recognise them quickly, even from the back. The unique 8-sided shape is different from all other road signs, making it easy to identify even without reading the word STOP.
Q8. What is tessellation?
Tessellation is covering a flat surface with polygons so there are no gaps or overlaps. Only certain polygons can tessellate by themselves: equilateral triangles, squares, and regular hexagons.
