## Learning Math: Geometry

# Symmetry Part B: Rotation Symmetry (30 minutes)

## Session 7, Part B

### In this part

- Determining Rotation Symmetry
- Creating Rotation Symmetry

### Determining Rotation Symmetry

If you can rotate (or turn) a figure around a center point by fewer than 360° and the figure appears unchanged, then the figure has rotation symmetry. The point around which you rotate is called the center of rotation, and the smallest angle you need to turn is called the angle of rotation.

This figure has rotation symmetry of 72°, and the center of rotation is the center of the figure:

**Problem B1**

a. |
Each of these figures has rotation symmetry. Can you estimate the center of rotation and the angle of rotation? |

b. |
Do the regular polygons have rotation symmetry? For each polygon, what are the center and angle of rotation? |

**Note 2**

As you will see in the next section, in order to have rotation symmetry, the center of rotation does not have to be the center of the figure. A figure can have rotation symmetry about a point that lies outside the figure.

Selected diagrams in Part B: Determining Rotation Symmetry taken from IMPACT Mathematics, Course 3, developed by Educational Development Center, Inc. p. 302. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

### Creating Rotation Symmetry

To create a design with rotation symmetry, you need three things:

- a figure, called a basic design element, that you will rotate
- a center of rotation
- an angle of rotation

By convention, we rotate figures counterclockwise for positive angles and clockwise for negative angles.

80° rotation about point P | -80° rotation about point P |

To create a symmetric design, follow the steps below:

**Step 1:** Copy this picture, including point P and the reference line. Or print the PDF version.

**Step 2:** Draw a new segment with P as one endpoint and forming a 60° angle with the reference line. Label this segment 1. Then draw four more segments, labeling them 2 through 5, from point P, each forming a 60° angle with the previous segment, as shown here:

**Step 3:** Place a sheet of tracing paper over your figure. Pin the papers together through the center of rotation. Trace the figure, including the reference line, but don’t trace segments 1-5.

**Step 4:** Now rotate your tracing until the reference line on the tracing is directly over segment 1. Trace the original figure again. Your tracing should now look like this.

**Step 5:** Rotate the tracing until the reference line on the tracing is directly over segment 2. Trace the original figure again.

**Step 6:** Repeat the process, rotating to place the reference line over the next segment and tracing the figure. Do this until the reference line on the tracing is back on the original reference line.

**Problem B2**

Does your design have reflection symmetry? If so, where is the line of symmetry?

**Problem B3**

Use the basic design element below and the given center of rotation to create a symmetric design with an angle of rotation of 120°. Does this design have reflection symmetry? If so, where is the line of symmetry? Printable PDF of design

Rotation Symmetry Steps 1-6 and Problems B2 and B3 adapted from IMPACT Mathematics, Course 3, developed by Educational Development Center, Inc. pp. 305-306. © 2000 Glencoe/McGraw-Hill. Used with permission www.glencoe.com/sec/math

### Notes

**Note 2
**

Note that the angle of rotation, which is equal to the external angle, is also equal to the central angle of the polygon. For regular polygons, the central angle has its vertex at the center of the polygon, and its rays go through any two adjacent vertices. In Session 4, we defined central angles in terms of circles. Here, you can think of circumscribing a circle about the regular polygon. Then these two notions of central angle coincide.

### Solutions

**Problem B1
**

a. |
For each shape, the center of rotation is the center of the figure. The angles of rotation, from left to right, are 120°, 180°, 120°, and 90°. |

b. |
Regular polygons do have rotation symmetry. In each case, the center of rotation is the center of the polygon, and the angle of rotation is 360°/n, where n is the number of sides in the polygon. So the angles are 120°, 90°, 72°, and 60°, respectively, for the equilateral triangle, square, regular pentagon, and regular hexagon. |

**Problem B2**

The design has no reflection symmetry.

**Problem B3**

The final design has three lines of symmetry. Each line of symmetry goes through the center of the rotation symmetry and is a perpendicular bisector of the straight line on the basic design element. Notice that the line symmetry arose because of line symmetry in the original basic design element, something that the first picture lacked.