Centre of Rotation
When a shape has rotational symmetry, it turns around a fixed point called the centre of rotation. Every point of the shape moves in a circle around this centre. The centre itself does not move.
For regular shapes, the centre of rotation is the point where all axes of symmetry meet — the geometric centre of the shape.
What is Centre of Rotation - Grade 7 Maths (Symmetry)?
Definition: The centre of rotation is the fixed point around which a figure rotates. All other points of the figure move in circular paths around this point.
- For a square: the centre is the intersection of its diagonals.
- For a circle: the centre of the circle itself.
- For an equilateral triangle: the centroid (where medians meet).
Centre of Rotation Formula
Locating the centre of rotation:
- For regular polygons: the centre is equidistant from all vertices.
- For shapes with line symmetry: the centre is where the axes of symmetry intersect.
- For a circle: the centre of the circle.
Types and Properties
Centre of rotation for common shapes:
- Square: Intersection of diagonals.
- Rectangle: Intersection of diagonals.
- Equilateral triangle: Centroid (where all medians meet).
- Regular hexagon: Centre point equidistant from all 6 vertices.
- Circle: Its own centre.
- Parallelogram: Intersection of diagonals.
Solved Examples
Example 1: Centre of a Square
Problem: Where is the centre of rotation of a square ABCD?
Solution:
- Draw diagonals AC and BD.
- The point where they cross is the centre of rotation.
Answer: The centre is the intersection of the diagonals.
Example 2: Centre of an Equilateral Triangle
Problem: Find the centre of rotation of an equilateral triangle.
Solution:
- Draw all three medians (line from vertex to midpoint of opposite side).
- They all meet at one point — the centroid.
Answer: The centre is the centroid.
Example 3: Centre of a Circle
Problem: What is the centre of rotation of a circle?
Solution:
- A circle looks the same after any rotation about its centre.
Answer: The centre of rotation is the centre of the circle.
Example 4: Verifying the Centre
Problem: A regular pentagon is rotated about a point and it maps onto itself. Where is this point?
Solution:
- The centre is equidistant from all 5 vertices.
- It is the point where all lines of symmetry meet.
Answer: The geometric centre of the pentagon.
Real-World Applications
Real-world uses:
- Wheels and axles: The axle is the centre of rotation of a wheel.
- Clock hands: The pin at the centre of a clock is the centre of rotation.
- Fans and turbines: Blades rotate about a central hub.
- Doors: The hinge acts as the centre of rotation.
Key Points to Remember
- The centre of rotation is the fixed point that does not move during rotation.
- All other points move in circles around the centre.
- For regular polygons, the centre is equidistant from all vertices.
- The centre lies at the intersection of lines/axes of symmetry.
- A shape can be rotated clockwise or anticlockwise about its centre.
Practice Problems
- Where is the centre of rotation of a regular hexagon?
- A door rotates on its hinge. Is the hinge the centre of rotation?
- Find the centre of rotation of a rectangle.
- Does a scalene triangle have a centre of rotation that gives rotational symmetry of order 2 or more?
Frequently Asked Questions
Q1. What is the centre of rotation?
It is the fixed point around which a shape rotates. All other points of the shape move in circular paths around it.
Q2. How do you find the centre of rotation of a regular polygon?
Draw the diagonals or axes of symmetry. The point where they all meet is the centre of rotation.
Q3. Does every shape have a centre of rotation?
Every shape can be rotated about any point, but it only has rotational symmetry if it maps onto itself. Shapes without rotational symmetry (order 1 only) do not have a useful centre.
