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

Learn more about rotational symmetry by watching the video at this link.

### Guidance

**Rotational Symmetry** is when a figure can be rotated (less that ) and it looks the same way it did before the rotation. The **center of rotation** is the point at which 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, figures will have an **angle of rotation**, that tells us how many degrees we can rotate a figure so that it still looks the same. In general, if a shape can be rotated n times, the angle of rotation is . Then, multiply the angle of rotation by 1, 2, 3..., and to find the additional angles of rotation.

#### Example A

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

The pentagon can be rotated 4 times and show rotational symmetry. Because there are 5 lines of rotational symmetry, the angle would be . Note that the 5th rotation would be and so does not count for demonstrating rotational symmetry.

#### Example B

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

The can be rotated once. The angle of rotation is .

#### 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 3 times. There are 4 lines of rotational symmetry, so the angle of rotation is . It can also be rotated and and it will still look the same.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter12RotationSymmetryB

#### Concept Problem Revisited

The starfish has rotational symmetry of . Therefore, the starfish can be rotated , and 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. The angle of rotation is .

2. The hexagon can be rotated six times. The angle of rotation is .

3. This figure can be rotated four times. The angle of rotation is .

### 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 times, it has _______ lines of rotational symmetry.
- To find the angle of rotation, divide 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.