### Rotation Symmetry

A figure exhibits** rotational symmetry** if it can be rotated (less than \begin{align*}360^\circ\end{align*}) and look the same as it did before the rotation. The **center of rotation** is the point that the figure is rotated around such that the rotational symmetry holds. Typically, the center of rotation is the center of the figure. Along with rotational symmetry and a center of rotation is the **angle of rotation** that tells us how many degrees to rotate the figure so that it still looks the same. In general, if a shape can be rotated n times, the angle of rotation is \begin{align*}\frac{360^\circ}{n}.\end{align*} Multiply the angle of rotation by 1, 2, 3...\begin{align*}n\end{align*} to find the additional angles of rotation. The number of rotations may be referred to as the **order**.

#### Recognizing Rotational Symmetry

1. Determine if the figure below has rotational symmetry. Find the angle of rotational symmetry and the order.

A regular pentagon can be rotated 5 times to demonstrate rotational symmetry, so it is order 5. Because there are 5 lines of rotational symmetry, the angle would be \begin{align*}\frac{360^\circ}{5}= 72^\circ.\end{align*}

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

The \begin{align*}N\end{align*} can be rotated twice, so it is order 2. The angle of rotation is \begin{align*}180^\circ.\end{align*}

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

The checkerboard can be rotated 4 times. There are 4 lines of rotational symmetry, so the angle of rotation is \begin{align*}\frac{360^\circ}{4}=90^\circ.\end{align*} It appears the same at \begin{align*}0^\circ (\text{or 360}^\circ), 90^\circ, 180^\circ, \text{ and }270^\circ.\end{align*}

#### Starfish Problem Revisited

The starfish has rotational symmetry of \begin{align*}72^\circ\end{align*}. Therefore, the starfish can be rotated \begin{align*}72^\circ, 144^\circ, 216^\circ, 288^\circ,\end{align*} and \begin{align*}360^\circ\end{align*} and it will still look the same. The center of rotation is the center of the starfish.

### Examples

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

#### Example 1

The parallelogram can be rotated twice, so it is order 2. The angle of rotation is \begin{align*}180^\circ.\end{align*}

#### Example 2

The hexagon can be rotated six times, order 6. The angle of rotation is \begin{align*}60^\circ.\end{align*}

#### Example 3

This figure can be rotated four times, order 4. The angle of rotation is \begin{align*}90^\circ.\end{align*}

### 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 \begin{align*}n\end{align*} times, it has _______ lines of rotational symmetry.
- To find the angle of rotation, divide \begin{align*}360^\circ\end{align*} 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 view the Review answers, open this PDF file and look for section 12.2.