Symmetry in Plane Figures
Understanding balance and regularity in shapes
What is Symmetry?
Symmetry occurs when a shape can be transformed (by reflection, rotation, or translation) and still look exactly the same as before.
Symmetry is everywhere in nature, art, and mathematics — from butterfly wings to snowflakes to geometric shapes.
Types of Symmetry
Line Symmetry (Reflective)
A shape has line symmetry if it can be folded along a line so that both halves match exactly.
The fold line is called the axis of symmetry or mirror line.
Rotational Symmetry
A shape has rotational symmetry if it looks the same after being rotated less than 360°.
The order tells how many times it matches in one full rotation.
Line Symmetry Examples
Square
4 lines of symmetry
Rectangle
2 lines of symmetry
Equilateral Triangle
3 lines of symmetry
Isosceles Triangle
1 line of symmetry
Circle
Infinite lines
Parallelogram
0 lines of symmetry
Rotational Symmetry
Order of Rotational Symmetry
The order of rotational symmetry is the number of times a shape fits onto itself during a full 360° rotation.
To find the angle of rotation: Angle = 360° ÷ Order
A shape with order 1 has no rotational symmetry (it only matches at 360°, back where it started).
Examples of Rotational Symmetry
| Shape | Order | Angle of Rotation |
|---|---|---|
| Equilateral Triangle | 3 | 120° |
| Square | 4 | 90° |
| Regular Pentagon | 5 | 72° |
| Regular Hexagon | 6 | 60° |
| Circle | Infinite | Any angle |
Interactive Symmetry Explorer
Explore Symmetry
Equilateral Triangle: 3 lines of symmetry, Order 3 rotational symmetry (120°)
Regular Polygons and Symmetry
Pattern for Regular Polygons
For a regular polygon with n sides:
- Lines of symmetry = n
- Order of rotational symmetry = n
- Angle of rotation = 360° ÷ n
This makes regular polygons particularly "symmetric" — they have the maximum symmetry for their number of sides!
Practice Problems
Question 1: How many lines of symmetry does a regular octagon have?
Show Solution
A regular octagon has 8 sides, so it has 8 lines of symmetry.
Question 2: What is the order of rotational symmetry of a regular hexagon, and what is the angle of rotation?
Show Solution
Order = 6 (matches itself 6 times in 360°)
Angle = 360° ÷ 6 = 60°
Question 3: A rhombus has how many lines of symmetry?
Show Solution
A rhombus has 2 lines of symmetry — they are its diagonals.
Question 4: Does a scalene triangle have any symmetry?
Show Solution
A scalene triangle has:
- 0 lines of symmetry
- Order 1 rotational symmetry (no rotational symmetry)
Question 5: The letter "H" has how many lines of symmetry?
Show Solution
The letter H has 2 lines of symmetry: one vertical and one horizontal through the center.
CSEC Exam Tips
- Draw the lines of symmetry clearly with a ruler
- For rotational symmetry, trace the shape and rotate to check
- Remember: regular n-gon = n lines = order n
- A shape can have line symmetry, rotational symmetry, both, or neither