Two-Dimensional Figures - Sixth Grade
Complete Notes & Formulas
1. Polygons - Definition and Classification
What is a Polygon?
A polygon is a CLOSED 2D shape made of straight lines
• Must have at least 3 sides
• All sides are line segments (straight lines)
• Must be closed (all sides connect)
• No curved sides
Polygon Names by Number of Sides
| Number of Sides | Polygon Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon (Septagon) |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
Regular vs Irregular Polygons
| Type | Definition | Example |
|---|---|---|
| Regular Polygon | All sides equal AND all angles equal | Equilateral triangle, Square |
| Irregular Polygon | Sides OR angles are NOT all equal | Rectangle, Scalene triangle |
2. Classifying Triangles
Classification by SIDES
| Type | Sides | Description |
|---|---|---|
| Equilateral | 3 equal sides | All sides same length, all angles 60° |
| Isosceles | 2 equal sides | Two sides same length, two angles equal |
| Scalene | 0 equal sides | All sides different lengths, all angles different |
Classification by ANGLES
| Type | Angles | Description |
|---|---|---|
| Acute Triangle | All angles < 90° | All three angles are acute |
| Right Triangle | One angle = 90° | Has one right angle (L shape) |
| Obtuse Triangle | One angle > 90° | Has one obtuse angle (wide angle) |
Important: A triangle can be classified by BOTH sides AND angles. Example: "Right Isosceles Triangle"
3. Triangle Inequality Theorem
The sum of any TWO sides of a triangle
must be GREATER THAN the third side
Formula
For a triangle with sides a, b, and c:
a + b > c
b + c > a
a + c > b
ALL three conditions must be true!
Example: Can sides 3, 4, 5 form a triangle?
Check all three conditions:
3 + 4 > 5 → 7 > 5 ✓ (TRUE)
4 + 5 > 3 → 9 > 3 ✓ (TRUE)
3 + 5 > 4 → 8 > 4 ✓ (TRUE)
Answer: YES, these sides can form a triangle!
Example: Can sides 2, 3, 7 form a triangle?
Check:
2 + 3 > 7 → 5 > 7 ✗ (FALSE)
Answer: NO! These sides CANNOT form a triangle!
4. Classifying Quadrilaterals
What is a Quadrilateral?
• A polygon with FOUR sides
• Has FOUR vertices (corners)
• Sum of all interior angles = 360°
Types of Quadrilaterals
| Type | Properties |
|---|---|
| Trapezoid (Trapezium) | ONE pair of parallel sides |
| Parallelogram | TWO pairs of parallel sides, opposite sides equal |
| Rectangle | Parallelogram with 4 right angles (90°) |
| Rhombus | Parallelogram with 4 equal sides |
| Square | Rectangle AND Rhombus (4 equal sides, 4 right angles) |
| Kite | Two pairs of adjacent sides equal |
Hierarchy: Square → Rectangle → Parallelogram → Quadrilateral. A square is ALL of these!
5. Sum of Interior Angles in Polygons
The Formula
Sum of Interior Angles
S = (n − 2) × 180°
where n = number of sides
Sum of Angles for Common Polygons
| Polygon | Sides (n) | Formula | Sum |
|---|---|---|---|
| Triangle | 3 | (3−2) × 180° | 180° |
| Quadrilateral | 4 | (4−2) × 180° | 360° |
| Pentagon | 5 | (5−2) × 180° | 540° |
| Hexagon | 6 | (6−2) × 180° | 720° |
| Octagon | 8 | (8−2) × 180° | 1080° |
Each Interior Angle of Regular Polygon
Each angle = [(n − 2) × 180°] ÷ n
6. Finding Missing Angles in Triangles
Triangle Angle Sum Theorem
∠A + ∠B + ∠C = 180°
The sum of all angles in a triangle is ALWAYS 180°
Example: Find the missing angle
Problem: A triangle has angles 50° and 70°. Find the third angle.
∠A + ∠B + ∠C = 180°
50° + 70° + ∠C = 180°
120° + ∠C = 180°
∠C = 180° − 120°
∠C = 60°
Answer: The third angle is 60°
Special Triangles
Equilateral Triangle: All angles = 60°
Isosceles Triangle: Two angles are equal
Right Triangle: One angle = 90°, other two angles add to 90°
7. Finding Missing Angles in Quadrilaterals
Quadrilateral Angle Sum
∠A + ∠B + ∠C + ∠D = 360°
The sum of all angles in a quadrilateral is ALWAYS 360°
Example
Problem: A quadrilateral has angles 80°, 110°, and 70°. Find the fourth angle.
80° + 110° + 70° + ∠D = 360°
260° + ∠D = 360°
∠D = 360° − 260°
∠D = 100°
Answer: The fourth angle is 100°
8. Line Symmetry
What is Line Symmetry?
A shape has line symmetry when it can be
folded along a line and BOTH halves match perfectly
The line is called the LINE OF SYMMETRY or AXIS OF SYMMETRY
Lines of Symmetry in Shapes
| Shape | Number of Lines of Symmetry |
|---|---|
| Circle | Infinite (∞) |
| Equilateral Triangle | 3 |
| Isosceles Triangle | 1 |
| Scalene Triangle | 0 |
| Square | 4 |
| Rectangle | 2 |
| Rhombus | 2 |
| Regular Pentagon | 5 |
| Regular Hexagon | 6 |
Pattern: A regular polygon with n sides has n lines of symmetry!
Quick Reference: Key Formulas
| Concept | Formula |
|---|---|
| Triangle Angle Sum | ∠A + ∠B + ∠C = 180° |
| Quadrilateral Angle Sum | Sum = 360° |
| Polygon Angle Sum | S = (n − 2) × 180° |
| Triangle Inequality | a + b > c |
| Regular Polygon Each Angle | [(n − 2) × 180°] ÷ n |
💡 Important Tips to Remember
✓ Triangle angles always add to 180°
✓ Quadrilateral angles always add to 360°
✓ Triangle Inequality: Sum of any two sides > third side
✓ Regular polygon: All sides AND all angles equal
✓ Square is a special rectangle AND rhombus
✓ Equilateral triangle: 3 equal sides, all angles 60°
✓ Isosceles triangle: 2 equal sides, 2 equal angles
✓ Trapezoid: Exactly ONE pair of parallel sides
✓ Line of symmetry: Divides shape into mirror halves
✓ Use (n−2) × 180° for ANY polygon angle sum
🧠 Memory Tricks & Strategies
Triangle Angles:
"TRI-angle has THREE angles that make 180!"
Quadrilateral Angles:
"QUAD means FOUR - 4 sides, 4 angles, and they add to 360 (like a full circle)!"
Triangle Inequality:
"Two short sides together must be LONGER than the long side!"
Regular vs Irregular:
"REGULAR means EQUAL - all sides equal, all angles equal!"
Polygon Angle Sum:
"Take sides MINUS TWO, times 180 - that's the angle sum for sure!"
Line Symmetry:
"If you can FOLD it and MATCH it, it has symmetry - just catch it!"
Master Two-Dimensional Figures! 🔺 ⬜ ⬟ ⬡
Remember: Practice identifying and classifying shapes!