Rotational Symmetry

Rotational Symmetry of various geometric shapes tells how many times a shape aligns to its original position when it is rotated 360 degrees. Various figures having rotational symmetry are Square, Circle, Rectangle, Equilateral Triangle, and others.

Symmetry refers to the balanced likeness and proportion between two halves of an object, where one side mirrors the other. Conversely, asymmetry denotes a lack of this balance. Symmetry manifests in nature, architecture, and art, and can be observed through flipping, sliding, or rotating objects. Different types of symmetry include :

• Reflection
• Translational
• Rotational

Rotational Symmetry Definition

Rotational symmetry is observed in shapes or figures that retain their appearance even after being rotated around a specific central point. Imagine a shape like a square or a circle. If you were to rotate it around its center it would look identical at specific intervals of rotation (like after a quarter turn for a square or after any degree of rotation for a circle). This characteristic defines rotational symmetry.

Shapes exhibiting this property are commonly found in geometry. For instance squares, circles, and regular polygons (such as hexagons) are classic examples.

Examples of Rotational Symmetry

Rotational Symmetry of various figures are added in the article below,

Rotational Symmetry of a Parallelogram

A parallelogram may demonstrate rotational symmetry if it can be rotated about its center by a certain angle and still maintain its original appearance. This property is often evident in parallelograms with congruent angles and side lengths, such as rectangles or rhombuses.

Rotational Symmetry of a Rectangle

A rectangle possesses rotational symmetry of order 2. This means it aligns with its original position after being rotated by 180 degrees around its center due to its equal side lengths and congruent angles.

Rotational Symmetry of a Square

A square displays rotational symmetry of order 4. It aligns perfectly with its original position after being rotated by 90 degrees successively four times around its center due to its equal side lengths and congruent angles.

Order of Rotational Symmetry of Square

The order of rotational symmetry in a square is 4, implying it has four positions (90, 180, 270, and 360 degrees) where it coincides with its initial orientation.

Rotational Symmetry of a Rhombus

A rhombus typically has rotational symmetry of order 2 but may possess higher-order symmetry depending on its angles. It can align with its original position after a 180-degree rotation around its center.

Rotational Symmetry of a Pentagon

A regular pentagon exhibits rotational symmetry of order 5. It can be rotated by 72 degrees successively five times about its center and still coincide with its initial orientation.

Rotational Symmetry of a Hexagon

A regular hexagon demonstrates rotational symmetry of order 6. It can be rotated by 60 degrees successively six times about its center and maintain its original appearance.

Rotational Symmetry of an Equilateral Triangle

An equilateral triangle shows rotational symmetry of order 3. It aligns with its original position after being rotated by 120 degrees successively three times around its center due to its congruent sides and angles.

Triangle Rotational Symmetry

Rotational symmetry in triangles varies by type. Equilateral triangles possess rotational symmetry due to their equal sides and angles, while isosceles and scalene triangles typically lack this property. Equilateral triangles specifically demonstrate rotational symmetry of order 3, aligning with their original position after a 120-degree rotation.

Center of Rotation

Center of Rotation refers to a fixed point around which a shape or object rotates. When you perform a rotational transformation, every point in the figure moves in a circular path around this central point by a specific angle. This point remains stationary while the rest of the object moves in a circular motion around it. It’s akin to the pivot or axis point for the rotation, defining the point of reference around which the figure revolves.

Angle of Rotational Symmetry

The angle of rotational symmetry refers to the smallest angle through which a shape can be rotated while retaining its original appearance. It represents the minimum angle required to bring the shape back to its initial orientation through repeated rotations.

For example, if a shape aligns perfectly after a 120° rotation then 120° is its angle of rotational symmetry. This angle signifies how the shape repeats its appearance under rotation.

Order of Rotational Symmetry

The order of rotational symmetry denotes how many times a shape aligns with its original position during a full 360-degree rotation. It signifies the number of positions in which a shape appears identical to its initial orientation as it’s rotated around its center.

For example, if a shape aligns twice during a complete revolution, it has an order of rotational symmetry of 2. Shapes with higher orders of rotational symmetry match their original positions more times within a full rotation, showcasing more symmetry under rotation.

Rotational Symmetry Letters

Rotational symmetry in letters refers to certain alphabet characters that possess symmetry when rotated around a central point. Some letters, such as “O,” “X,” “H,” and “I,” exhibit rotational symmetry.

For instance, the letter “O” maintains its appearance when rotated by 180 degrees, “X” aligns at 180 degrees, “H” remains the same when rotated by 180 degrees, and “I” retains its form at 180 degrees. These letters can be rotated by specific angles and still resemble their original shapes. However, many other letters, like “A,” “B,” “C,” lack this property as they cannot be rotated to match their original forms within 360 degrees of rotation.

Solved Examples on Rotational Symmetry

Example 1: Calculate the order of rotational symmetry for a regular hexagon?

Solution:

A regular hexagon, with its 6 equal sides, demonstrates rotational symmetry. It aligns perfectly after successive rotations of 60 degrees, completing a full 360-degree rotation.

Therefore, the order of rotational symmetry for a regular hexagon is 6.

Example 2: Find the angle of rotational symmetry for an equilateral triangle?

Solution:

An equilateral triangle, having 3 equal sides, exhibits rotational symmetry. It aligns perfectly after a 120-degree rotation around its center.

Hence, the angle of rotational symmetry for an equilateral triangle is 120 degrees.

Example 3: Determine the order of rotational symmetry for a square?

Solution:

A square, featuring 4 equal sides, can be rotated by 90 degrees successively four times around its center, completing a 360-degree rotation and aligning with its original position.

Therefore, the order of rotational symmetry for a square is 4.

Example 4: Calculate the angle of rotational symmetry for a regular pentagon?

Solution:

A regular pentagon, with 5 equal sides, demonstrates rotational symmetry. It aligns perfectly after a 72-degree rotation around its center. Hence, the angle of rotational symmetry for a regular pentagon is 72 degrees.

Practice Problems on Rotational Symmetry

Problem 1: Identify shapes exhibiting rotational symmetry and specify the degrees of rotation maintaining their original appearance.

1. Pentagon
2. Rectangle
3. Regular Octagon
4. Parallelogram

Problem 2: Sketch a shape possessing exactly two lines of rotational symmetry?

Problem 3: Determine the angle of rotation for a regular heptagon (7-sided polygon) to retain its original appearance?

Problem 4: Identify English alphabet letters displaying rotational symmetry. List the identified letters?

Problem 5: Given an irregular shape, ascertain if it demonstrates any rotational symmetry. Describe the degrees of rotation where it maintains its original appearance if applicable?

Problem 6: Illustrate a shape exhibiting more than eight lines of rotational symmetry?

Problem 7: For a shape demonstrating rotational symmetry, with an angle of rotation set at 45 degrees calculate the number of positions it occupies in a complete rotation (360 degrees)?

Problem 8: Determine the validity of the statement: “All regular polygons possess rotational symmetry”?

Problem 9: Construct a shape showcasing both rotational and reflectional symmetry?

Problem 10: Find the minimum number of lines of symmetry necessary for a shape to also possess rotational symmetry?

Rotational Symmetry – FAQs

1. What is Rotational Symmetry?

Rotational symmetry refers to the property of a shape or figure that remains unchanged in appearance after being rotated around a fixed central point. When the figure is rotated by a certain angle (usually less than 360 degrees), it aligns perfectly with its original form.

2. What is Difference Between Rotational Symmetry and Line Symmetry?

Rotational symmetry involves a figure rotating around a central point and retaining its appearance at specific intervals of rotation. Line symmetry (or reflectional symmetry) occurs when a figure reflects over a line and its halves match perfectly.

3. Can a Shape have both Rotational and Line Symmetry?

Yes, some shapes possess both rotational and line symmetry. For instance, a regular hexagon has both Rotational Symmetry and Line Symmetry.

4. What is Center of Rotation?

Center of Rotation is a fixed point around which an object rotates during a transformation.

5. What is Angle of Rotational Symmetry?

Angle of Rotational Symmetry refers to the smallest angle by which a shape or figure can be rotated and still coincide with its original position.

6. What is Order of Rotational Symmetry?

Order of Rotational Symmetry denotes how many times a shape aligns with its original position during a full 360-degree rotation.

7. Which Letters Exhibit Rotational Symmetry?

Various letters that exhibit Rotational Symmetry are Z, H, S, N and O.