Rotational Symmetry
A shape has rotational symmetry if it looks exactly the same after being rotated (turned) by some angle less than 360° about its centre. For example, a square looks the same after a 90° turn.
The number of times a shape fits onto itself during a full 360° rotation is called the order of rotational symmetry.
What is Rotational Symmetry - Grade 7 Maths (Symmetry)?
Definition: A figure has rotational symmetry if it maps onto itself after rotation through an angle less than 360° about a fixed point (centre).
- Order of rotation = number of times the shape looks the same during a full turn.
- Angle of rotation = 360° ÷ order.
Rotational Symmetry Formula
Angle of Rotation:
Angle of rotation = 360° ÷ Order
Orders for common shapes:
- Equilateral triangle: order 3, angle = 120°
- Square: order 4, angle = 90°
- Regular pentagon: order 5, angle = 72°
- Regular hexagon: order 6, angle = 60°
- Circle: order infinite
Types and Properties
Shapes and their rotational symmetry:
- No rotational symmetry (order 1): Scalene triangle, letter R, letter F.
- Order 2: Rectangle, parallelogram, letter S, letter Z.
- Order 3: Equilateral triangle, recycling symbol.
- Order 4: Square, plus sign (+).
- Order 6: Regular hexagon, Star of David.
Solved Examples
Example 1: Order of a Square
Problem: What is the order of rotational symmetry of a square?
Solution:
- A square looks the same at 90°, 180°, 270°, and 360°.
- It fits onto itself 4 times in a full turn.
Answer: Order = 4, angle of rotation = 90°.
Example 2: Order of an Equilateral Triangle
Problem: Find the order of rotational symmetry of an equilateral triangle.
Solution:
- An equilateral triangle looks the same at 120°, 240°, and 360°.
- It fits onto itself 3 times.
Answer: Order = 3, angle = 120°.
Example 3: Rectangle vs Square
Problem: What is the order of rotational symmetry of a rectangle (that is not a square)?
Solution:
- A rectangle looks the same at 180° and 360°.
- It fits onto itself 2 times.
Answer: Order = 2, angle = 180°.
Example 4: Finding Angle from Order
Problem: A regular polygon has rotational symmetry of order 8. Find the angle of rotation.
Solution:
- Angle = 360° ÷ 8 = 45°
Answer: 45°.
Real-World Applications
Real-world uses:
- Design: Logos, patterns, and floor tiles use rotational symmetry.
- Nature: Flowers, starfish, and snowflakes show rotational symmetry.
- Engineering: Wheels, gears, and propellers rely on rotational symmetry for balance.
Key Points to Remember
- A shape has rotational symmetry if it looks the same after rotation by less than 360°.
- Order = number of times it matches during a full 360° turn.
- Angle of rotation = 360° ÷ order.
- Every shape has at least order 1 (360° rotation).
- A circle has infinite order of rotational symmetry.
- Regular polygons with n sides have rotational symmetry of order n.
Practice Problems
- What is the order of rotational symmetry of a regular hexagon?
- Find the angle of rotation for a shape with order 5.
- Does a parallelogram have rotational symmetry? If yes, what order?
- Name two letters of the alphabet with rotational symmetry of order 2.
Frequently Asked Questions
Q1. What is rotational symmetry?
A figure has rotational symmetry if it looks the same after being turned by some angle less than 360° about its centre.
Q2. How do you find the order of rotational symmetry?
Rotate the shape by equal angles (360° ÷ n) and count how many times it looks the same in a complete turn. That count is the order.
Q3. Does every shape have rotational symmetry?
Every shape has order 1 (it matches at 360°). But we say a shape has rotational symmetry only if the order is 2 or more.
