# 8.4: Rotation Symmetry

- Page ID
- 2163

Rotation less than 360 degrees that carries a shape onto itself.

**Rotational symmetry** is present when a figure can be rotated (less than \(360^{\circ}\)) such that it looks like it did before the **rotation**. The **center of rotation** is the point a figure is rotated around such that the rotational **symmetry** holds.

For the \(H\), we can rotate it twice, the triangle can be rotated 3 times and still look the same and the hexagon can be rotated 6 times.

What if you had a six-pointed star and you rotated that star less than \(360^{\circ}\)? If the rotated star looked exactly the same as the original star, what would that say about the star?

Example \(\PageIndex{1}\)

Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.

**Solution**

The pentagon can be rotated 5 times. Because there are 5 lines of rotational symmetry, the angle would be \(\dfrac{360^{\circ}}{5}=72^{\circ}\).

Example \(\PageIndex{2}\)

Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.

**Solution**

The N can be rotated twice. This means the angle of rotation is \(180^{\circ}\).

Example \(\PageIndex{3}\)

Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.

**Solution**

The checkerboard can be rotated 4 times. There are 4 lines of rotational symmetry, so the angle of rotation is \(\dfrac{360^{\circ}}{4}=90^{\circ}\).

Example \(\PageIndex{4}\)

Find the angle of rotation and the number of times each figure can rotate.

**Solution**

The parallelogram can be rotated twice. The angle of rotation is \(180^{\circ}\).

Example \(\PageIndex{5}\)

**Solution**

The hexagon can be rotated six times. The angle of rotation is \(60^{\circ}\).

## Review

- If a figure has 3 lines of rotational symmetry, it can be rotated _______ times.
- If a figure can be rotated 6 times, it has _______ lines of rotational symmetry.
- If a figure can be rotated n times, it has _______ lines of rotational symmetry.
- To find the angle of rotation, divide \(360^{\circ}\) by the total number of _____________.
- Every square has an angle of rotation of _________.

Determine whether each statement is true or false.

- Every parallelogram has rotational symmetry.
- Every figure that has line symmetry also has rotational symmetry.

Determine whether the words below have rotation symmetry.

**OHIO****MOW****WOW****KICK****pod**

Find the angle of rotation and the number of times each figure can rotate.

Determine if the figures below have rotation symmetry. Identify the angle of rotation.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 12.2.

## Vocabulary

Term | Definition |
---|---|

rotational symmetry |
When a figure can be rotated (less than 360^{\circ}\)) such that it looks like it did before the rotation. The is the point a figure is rotated around such that the rotational symmetry holds.center of rotation |

Center of Rotation |
In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point. |

Rotation |
A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure. |

Rotation Symmetry |
A figure has rotational symmetry if it can be rotated less than \(360^{\circ}\) around its center point and look exactly the same as it did before the rotation. |

Symmetry |
A figure has symmetry if it can be transformed and still look the same. |

## Additional Resources

Interactive Element

Video: Rotation Symmetry Principles - Basic

Activities: Rotation Symmetry Discussion Questions

Study Aids: Symmetry and Tessellations Study Guide

Practice: Rotation Symmetry

Real World: This End Up