Polygons - Fifth Grade
Complete Notes & Formulas
1. Is it a Polygon?
What is a Polygon?
A polygon is a closed, two-dimensional shape made up of straight line segments that connect end to end.
"Poly" = Many
"Gon" = Angles/Sides
Polygon = Many sides!
Requirements for a Polygon
1. Closed Shape
All sides must connect to form a complete figure (no openings)
2. Straight Sides Only
Made only of straight line segments (no curves or rounded edges)
3. At Least 3 Sides
Minimum of 3 sides (triangle is the simplest polygon)
4. Two-Dimensional (Flat)
A flat shape on a plane (not a 3D solid)
5. Simple (Non-Intersecting)
Sides don't cross over each other
What is NOT a Polygon?
✗ Circle - has a curved edge (no straight sides)
✗ Oval/Ellipse - curved shape
✗ Open shapes - sides don't connect completely
✗ Shapes with curved sides - like a heart or crescent
✗ 3D shapes - like cube, sphere, cone (these are polyhedrons, not polygons)
Examples
| Shape | Is it a Polygon? | Reason |
|---|---|---|
| Triangle | YES ✓ | Closed, 3 straight sides |
| Square | YES ✓ | Closed, 4 straight sides |
| Pentagon | YES ✓ | Closed, 5 straight sides |
| Circle | NO ✗ | Curved edge, no straight sides |
| Open figure (incomplete) | NO ✗ | Not closed |
Quick Test: Can you trace it without lifting your pencil AND without crossing any curves? If yes, it's a polygon!
2. Number of Sides in Polygons
Polygon Names by Number of Sides
Polygons are named based on the number of sides they have. Each name tells us exactly how many sides the polygon has!
Common Polygon Names (3-10 Sides)
| Number of Sides | Polygon Name | Number of Angles | Example |
|---|---|---|---|
| 3 | Triangle (Trigon) | 3 | Yield sign |
| 4 | Quadrilateral (Tetragon) | 4 | Square, rectangle |
| 5 | Pentagon | 5 | Pentagon building |
| 6 | Hexagon | 6 | Honeycomb cell |
| 7 | Heptagon (Septagon) | 7 | UK 50 pence coin |
| 8 | Octagon | 8 | Stop sign |
| 9 | Nonagon (Enneagon) | 9 | — |
| 10 | Decagon | 10 | — |
Extended List (11-20 Sides)
| 11 | Hendecagon (Undecagon) | 12 | Dodecagon |
| 13 | Triskaidecagon | 14 | Tetrakaidecagon |
| 15 | Pentadecagon | 16 | Hexakaidecagon |
| 17 | Heptadecagon | 18 | Octakaidecagon |
| 19 | Enneadecagon | 20 | Icosagon |
For Larger Polygons
For 13+ sides, you can simply say:
"n-gon" where n = number of sides
Example: 15-gon, 20-gon, 100-gon
Important Pattern
Key Rule: Number of Sides = Number of Angles = Number of Vertices
Example: A pentagon has 5 sides, 5 angles, and 5 vertices!
3. Regular and Irregular Polygons
A. Regular Polygons
Definition: A regular polygon has ALL sides equal in length AND ALL angles equal in measure.
Regular Polygon = Equilateral + Equiangular
All sides equal + All angles equal
Properties of Regular Polygons
• Equal sides: All sides have the same length
• Equal angles: All interior angles have the same measure
• Symmetrical: Have multiple lines of symmetry
• Can fit in a circle: All vertices touch a circle (circumscribed)
Examples of Regular Polygons
• Equilateral Triangle: 3 equal sides, 3 equal angles (60° each)
• Square: 4 equal sides, 4 equal angles (90° each)
• Regular Pentagon: 5 equal sides, 5 equal angles (108° each)
• Regular Hexagon: 6 equal sides, 6 equal angles (120° each)
• Regular Octagon: 8 equal sides, 8 equal angles (135° each)
Formula for Interior Angles of Regular Polygon
Each Interior Angle = (n - 2) × 180° ÷ n
where n = number of sides
B. Irregular Polygons
Definition: An irregular polygon does NOT have all sides equal OR does NOT have all angles equal (or both).
Irregular Polygon = Unequal sides OR Unequal angles
(or both)
Properties of Irregular Polygons
• Sides: At least one side has a different length
• Angles: At least one angle has a different measure
• Less symmetry: May have fewer or no lines of symmetry
• Still a polygon: Still closed with straight sides
Examples of Irregular Polygons
• Scalene Triangle: 3 unequal sides, 3 unequal angles
• Rectangle: 4 sides (2 pairs equal), but NOT all 4 equal
• Rhombus: 4 equal sides, but angles are NOT all equal
• Trapezoid: 4 unequal sides (except isosceles trapezoid)
• Irregular Pentagon: 5 sides that are not all equal
Comparison Table
| Property | Regular Polygon | Irregular Polygon |
|---|---|---|
| All sides equal | YES ✓ | NO ✗ |
| All angles equal | YES ✓ | NO ✗ |
| Symmetrical | Multiple lines | Few or none |
| Example | Square, Equilateral △ | Rectangle, Scalene △ |
Remember: A square is regular, but a rectangle is NOT regular (sides not all equal)!
4. Sort Polygons into Venn Diagrams
What is a Venn Diagram?
A Venn diagram uses overlapping circles to show relationships between different groups. It helps us see what properties polygons share!
How to Sort Polygons
Step 1: Look at the categories (labels on each circle)
Step 2: Check if the polygon fits ONE category
Step 3: Check if the polygon fits BOTH categories (overlap area)
Step 4: Place the polygon in the correct region
Common Venn Diagram Categories
Example 1: "Triangles" vs "Quadrilaterals"
• Circle 1 (Triangles): Triangle, equilateral triangle, isosceles triangle
• Circle 2 (Quadrilaterals): Square, rectangle, trapezoid
• Overlap: NONE (a shape cannot be both)
Example 2: "Regular Polygons" vs "4 Sides"
• Only Regular: Equilateral triangle, regular pentagon, regular hexagon
• Only 4 Sides: Rectangle, rhombus, trapezoid
• Overlap (Both): Square (regular AND 4 sides)
Example 3: "Right Angles" vs "All Sides Equal"
• Only Right Angles: Rectangle
• Only All Sides Equal: Rhombus, equilateral triangle
• Overlap (Both): Square
Practice Example
Venn Diagram: "Parallelograms" vs "4 Right Angles"
Left Circle Only (Parallelograms, no right angles):
• Rhombus (4 equal sides, not 90°)
• General parallelogram
Overlap (Both properties):
• Square (parallelogram with 4 right angles)
• Rectangle (parallelogram with 4 right angles)
Right Circle Only (4 right angles, not parallelogram):
• None! (All shapes with 4 right angles are parallelograms)
Tip: Overlap area is for shapes that have BOTH properties!
5. Properties of Polygons
General Polygon Properties
1. Sides
• Made of straight line segments
• Minimum of 3 sides
• Sides connect end-to-end
2. Angles
• Formed where two sides meet
• Number of angles = Number of sides
• Can be acute, right, or obtuse
3. Vertices (Corners)
• Points where sides meet
• Number of vertices = Number of sides
• Singular: vertex, Plural: vertices
4. Closed Figure
• All sides connect completely
• No gaps or openings
• Encloses a region
5. Two-Dimensional
• Flat shapes on a plane
• Have length and width (no depth)
Important Polygon Formulas
Sum of Interior Angles
(n - 2) × 180°
where n = number of sides
Examples:
• Triangle (3 sides): (3-2) × 180° = 180°
• Quadrilateral (4 sides): (4-2) × 180° = 360°
• Pentagon (5 sides): (5-2) × 180° = 540°
• Hexagon (6 sides): (6-2) × 180° = 720°
• Octagon (8 sides): (8-2) × 180° = 1080°
Other Important Formulas
Perimeter of Any Polygon:
P = sum of all side lengths
Perimeter of Regular Polygon:
P = n × s
where n = number of sides, s = length of one side
Number of Diagonals:
D = n(n - 3) ÷ 2
where n = number of sides
Properties Summary Table
| Polygon | Sides | Angles | Sum of Angles | Diagonals |
|---|---|---|---|---|
| Triangle | 3 | 3 | 180° | 0 |
| Quadrilateral | 4 | 4 | 360° | 2 |
| Pentagon | 5 | 5 | 540° | 5 |
| Hexagon | 6 | 6 | 720° | 9 |
| Octagon | 8 | 8 | 1080° | 20 |
💡 Important Tips to Remember
✓ Polygons must be closed and have straight sides
✓ Circles are NOT polygons (curved edge)
✓ Number of sides = Number of angles = Number of vertices
✓ Regular = All sides AND all angles equal
✓ Irregular = Sides OR angles are NOT all equal
✓ A square is regular, but a rectangle is irregular
✓ Sum of interior angles: (n - 2) × 180°
✓ For 13+ sides, use "n-gon" naming (15-gon, 20-gon)
✓ In Venn diagrams, overlap = shapes with BOTH properties
✓ "Poly" means many, "Gon" means angles
🧠 Memory Tricks
Polygon Names:
Triangle = Tricycle (3 wheels, 3 sides)
Quadrilateral = Quad bike (4 wheels, 4 sides)
Pentagon = Pentagon building has 5 sides
Hexagon = Hexagonal honeycomb (6 sides)
Octagon = Octopus (8 arms, 8 sides)
Regular vs Irregular:
Regular = "Regular" means the same every day - all equal!
Irregular = "Not regular" - things are different!
Is it a Polygon?
"Straight Sides, Closed Completely"
Sum of Angles Formula:
"Number of sides minus 2, then times 180"
Example: Pentagon → (5-2) × 180° = 540°
Master Polygons! ▲ ⬟ ⬢
Polygons are all around us - learn to identify and classify them!