What if you were asked to consider the presence of symmetry in nature? The starfish, below, is one example of symmetry in nature. Draw in the center of symmetry and the angle of rotation for this starfish. After completing this Concept, you'll be able to answer questions like these.

### Watch This

CK-12 Foundation: Chapter12RotationSymmetryA

### Guidance

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**.

#### Example A

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*}.

#### Example B

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*}

#### Example C

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*}

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter12RotationSymmetryB

#### Concept 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.

### Guided Practice

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

1.

2.

3.

**Answers:**

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

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

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

### Explore More

- 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.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.2.