Complete Notes on Polygons - Grade 5
1. What is a Polygon?
📐 Definition
A polygon is a closed figure made up of straight line segments (sides) that meet at points called vertices.
✓ Rules for a Shape to be a Polygon:
✅ It must be a CLOSED figure (starts and ends at the same point)
✅ It must have STRAIGHT SIDES only (no curved sides)
✅ It must have at least 3 or more sides
✅ It must be TWO-DIMENSIONAL (flat shape)
❌ NOT a Polygon if:
- The shape has curved sides (like a circle)
- The shape is open (doesn't close completely)
- The shape has less than 3 sides
- The sides cross over each other
✨ Examples:
Polygons: Triangle, Square, Rectangle, Pentagon, Hexagon, Octagon
NOT Polygons: Circle, Oval, Open shapes, Shapes with curved edges
2. Number of Sides in Polygons
Polygons are named based on the number of sides they have. The word "polygon" comes from Greek: "poly" means "many" and "gon" means "angle".
📊 Polygon Names by Number of Sides:
| Number of Sides | Polygon Name | Number of Angles |
|---|---|---|
| 3 | Triangle | 3 |
| 4 | Quadrilateral | 4 |
| 5 | Pentagon | 5 |
| 6 | Hexagon | 6 |
| 7 | Heptagon (Septagon) | 7 |
| 8 | Octagon | 8 |
| 9 | Nonagon | 9 |
| 10 | Decagon | 10 |
| 12 | Dodecagon | 12 |
| n | n-gon | n |
Important Rule: The number of sides = The number of angles = The number of vertices in any polygon
✨ Real-Life Examples:
- Triangle (3 sides): Pizza slice, road signs, musical instruments
- Quadrilateral (4 sides): Books, doors, windows, tables
- Pentagon (5 sides): The Pentagon building in USA, home plate in baseball
- Hexagon (6 sides): Honeycomb cells, nuts and bolts
- Octagon (8 sides): Stop signs
3. Regular and Irregular Polygons
🔷 Regular Polygons
A regular polygon is a polygon where:
- ALL sides have the SAME length
- ALL angles have the SAME measure
✨ Examples of Regular Polygons:
- Equilateral Triangle: All 3 sides equal, all 3 angles = \(60°\)
- Square: All 4 sides equal, all 4 angles = \(90°\)
- Regular Pentagon: All 5 sides equal, all 5 angles = \(108°\)
- Regular Hexagon: All 6 sides equal, all 6 angles = \(120°\)
- Regular Octagon: All 8 sides equal, all 8 angles = \(135°\)
🔶 Irregular Polygons
An irregular polygon is a polygon where:
- Sides have DIFFERENT lengths, OR
- Angles have DIFFERENT measures, OR
- BOTH sides and angles are different
✨ Examples of Irregular Polygons:
- Scalene Triangle: All 3 sides different lengths
- Rectangle: Opposite sides equal, but all 4 sides NOT equal
- Rhombus: All 4 sides equal, but angles are NOT all equal
- Trapezoid: Sides and angles are different
- Irregular Pentagon: Any pentagon with unequal sides or angles
📋 Quick Comparison Table:
| Feature | Regular Polygon | Irregular Polygon |
|---|---|---|
| Side Lengths | All equal | Not all equal |
| Angle Measures | All equal | Not all equal |
| Symmetry | High symmetry | May have low or no symmetry |
| Examples | Square, Equilateral triangle | Rectangle, Scalene triangle |
4. Sorting Polygons using Venn Diagrams
🔄 What is a Venn Diagram?
A Venn Diagram uses circles to show relationships between different groups. We can use Venn Diagrams to sort and classify polygons based on their properties.
📌 Common Ways to Sort Polygons:
1. By Number of Sides:
- 3 sides vs. 4 sides
- Less than 5 sides vs. 5 or more sides
2. By Regularity:
- Regular polygons vs. Irregular polygons
3. By Right Angles:
- Has right angles (90°) vs. No right angles
4. By Parallel Sides:
- Has parallel sides vs. No parallel sides
🎯 Example Sorting Categories:
Venn Diagram Example: Quadrilaterals with Right Angles AND Parallel Sides
- BOTH properties: Rectangle, Square
- Only Right Angles: Some trapezoids
- Only Parallel Sides: Parallelogram, Rhombus
- Neither: Irregular quadrilaterals
Venn Diagram Example: Regular Polygons AND Quadrilaterals
- BOTH: Square (only one!)
- Only Regular: Equilateral triangle, Regular pentagon, Regular hexagon
- Only Quadrilateral: Rectangle, Trapezoid, Rhombus
- Neither: Irregular triangles, Irregular pentagons
🔍 Tips for Sorting Polygons:
- Read the labels on each circle carefully
- Check ALL properties of each polygon
- If a polygon has BOTH properties, it goes in the overlapping section
- If it has NEITHER property, it goes outside both circles
5. Properties of Polygons
🔹 General Properties (All Polygons)
- Closed figure: Starts and ends at the same point
- Straight sides: Made only of line segments
- 2D shape: Flat, has length and width only
- Vertices (corners): Points where two sides meet
- Number of sides = Number of angles = Number of vertices
📐 Angle Properties
🔢 Sum of Interior Angles Formula
\[ \text{Sum of Interior Angles} = (n - 2) \times 180° \]
Where: \(n\) = number of sides
✨ Examples of Interior Angle Sums:
- Triangle (3 sides): \((3-2) \times 180° = 1 \times 180° = 180°\)
- Quadrilateral (4 sides): \((4-2) \times 180° = 2 \times 180° = 360°\)
- Pentagon (5 sides): \((5-2) \times 180° = 3 \times 180° = 540°\)
- Hexagon (6 sides): \((6-2) \times 180° = 4 \times 180° = 720°\)
- Octagon (8 sides): \((8-2) \times 180° = 6 \times 180° = 1080°\)
🔢 Each Interior Angle of a Regular Polygon
\[ \text{Each Interior Angle} = \frac{(n - 2) \times 180°}{n} \]
Where: \(n\) = number of sides
✨ Examples of Each Interior Angle (Regular Polygons):
- Equilateral Triangle: \(\frac{(3-2) \times 180°}{3} = \frac{180°}{3} = 60°\)
- Square: \(\frac{(4-2) \times 180°}{4} = \frac{360°}{4} = 90°\)
- Regular Pentagon: \(\frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°\)
- Regular Hexagon: \(\frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120°\)
- Regular Octagon: \(\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°\)
🔢 Sum of Exterior Angles
\[ \text{Sum of ALL Exterior Angles} = 360° \]
(This is TRUE for ANY polygon!)
🔢 Each Exterior Angle of a Regular Polygon
\[ \text{Each Exterior Angle} = \frac{360°}{n} \]
Where: \(n\) = number of sides
✨ Examples of Each Exterior Angle (Regular Polygons):
- Equilateral Triangle: \(\frac{360°}{3} = 120°\)
- Square: \(\frac{360°}{4} = 90°\)
- Regular Pentagon: \(\frac{360°}{5} = 72°\)
- Regular Hexagon: \(\frac{360°}{6} = 60°\)
- Regular Octagon: \(\frac{360°}{8} = 45°\)
📊 Summary Table of Common Regular Polygons
| Polygon | Sides | Sum of Interior Angles | Each Interior Angle | Each Exterior Angle |
|---|---|---|---|---|
| Equilateral Triangle | 3 | 180° | 60° | 120° |
| Square | 4 | 360° | 90° | 90° |
| Regular Pentagon | 5 | 540° | 108° | 72° |
| Regular Hexagon | 6 | 720° | 120° | 60° |
| Regular Heptagon | 7 | 900° | ≈128.57° | ≈51.43° |
| Regular Octagon | 8 | 1080° | 135° | 45° |
| Regular Nonagon | 9 | 1260° | 140° | 40° |
| Regular Decagon | 10 | 1440° | 144° | 36° |
🎓 Key Relationship: Interior Angle + Exterior Angle = 180° (they form a straight line!)
🎯 Additional Properties
Diagonal: A line segment connecting two non-adjacent vertices
🔢 Number of Diagonals Formula
\[ \text{Number of Diagonals} = \frac{n(n-3)}{2} \]
Where: \(n\) = number of sides
✨ Examples of Diagonals:
- Triangle: \(\frac{3(3-3)}{2} = 0\) diagonals
- Quadrilateral: \(\frac{4(4-3)}{2} = 2\) diagonals
- Pentagon: \(\frac{5(5-3)}{2} = 5\) diagonals
- Hexagon: \(\frac{6(6-3)}{2} = 9\) diagonals
📚 Quick Reference Summary
✓ Polygon Definition: Closed figure with straight sides, at least 3 sides
✓ Number Rule: Number of sides = Number of angles = Number of vertices
✓ Regular Polygon: All sides equal AND all angles equal
✓ Irregular Polygon: Sides OR angles are NOT all equal
✓ Interior Angles Sum: \((n-2) \times 180°\)
✓ Exterior Angles Sum: Always \(360°\)
✓ Most Common Polygons: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Octagon (8)