Rotational Symmetry — Definition, Formula & Examples
Rotational symmetry is a property of a shape that looks exactly the same after being rotated by some angle less than 360° around a central point. For example, a square has rotational symmetry because it matches its original position after a 90° turn.
A figure has rotational symmetry of order if there exist distinct rotations (including the full 360° rotation) about a fixed center point that map the figure onto itself. The smallest such positive angle of rotation is , called the angle of rotational symmetry.
Key Formula
Where:
- = The order of rotational symmetry (number of positions where the shape looks the same during a full rotation)
How It Works
To find a shape's rotational symmetry, pick its center and rotate the shape. Each time it looks identical to its starting position before completing a full turn, that counts as a symmetry. The total number of times it matches (including the full 360° rotation back to start) gives you the **order** of rotational symmetry. Divide 360° by the order to find the smallest angle of rotation. A shape with order 1 has no rotational symmetry — every shape maps onto itself after a full 360° turn, so that alone does not count.
Worked Example
Problem: What is the order of rotational symmetry and the angle of rotational symmetry for a regular hexagon?
Step 1: Count how many sides the regular hexagon has.
Step 2: For any regular polygon, the order of rotational symmetry equals the number of sides.
Step 3: Calculate the smallest angle of rotation by dividing 360° by the order.
Answer: A regular hexagon has rotational symmetry of order 6. It looks the same every 60° of rotation.
Another Example
Problem: Does the letter 'S' have rotational symmetry? If so, what is its order?
Step 1: Imagine rotating the letter S around its center. At 180°, the S looks identical to its original position.
Step 2: Check other angles. At 90° or 270°, the S does not match its original shape. The only match before 360° is at 180°.
Step 3: Count the total matches including 360°: the shape matches at 180° and 360°, giving order 2.
Answer: The letter S has rotational symmetry of order 2 with an angle of 180°.
Why It Matters
Rotational symmetry appears throughout middle-school geometry when classifying polygons and understanding transformations. Engineers and designers rely on it to create gears, turbines, and logos that look balanced from multiple viewing angles. In high-school courses like precalculus, recognizing rotational symmetry in graphs — such as the origin symmetry of odd functions — helps you sketch curves faster and verify equations.
Common Mistakes
Mistake: Counting the 360° rotation as proof of rotational symmetry.
Correction: Every shape returns to its original position after a 360° turn. A shape only has rotational symmetry if it matches at some angle less than 360°. Order 1 means no rotational symmetry.
Mistake: Confusing line symmetry with rotational symmetry.
Correction: Line (reflective) symmetry flips a shape across an axis. Rotational symmetry turns a shape around a point. A shape can have one type without the other, both, or neither.