# Rotational Symmetry

**K-12:**
Materials at high school level.

If points on a figure are equally positioned about a central point, then we say the object has **rotational symmetry**. A figure with rotational symmetry appears the same after rotating by some amount around the center point.

The **angle of rotation** of a symmetric figure is the smallest angle of rotation that preserves the figure. For example, the figure on the left can be turned by 180° (the same way you would turn an hourglass) and will look the same. The center (recycle) figure can be turned by 120°, and the star can be turned by 72°. For the star, where did 72° come from? The star has five points. To rotate it until it looks the same, you need to make <math>1/5</math> of a complete 360° turn. Since <math>1/5 \times 360^\circ = 72^\circ</math>, this is a 72° angle rotation.

Using degrees to describe the rotation amount is inconvenient because the precise angle is not obvious from looking at the figure. Instead, we will almost always use the **order of rotation** to describe rotational symmetry:

**Order of rotation**- A figure has order <math>n</math> rotational symmetry if <math>1/n</math> of a complete turn leaves the figure unchanged.

As with the star, you can compute the angle of rotation from the order of rotation:

An order <math>n</math> rotation corresponds to a <math>\frac{360^\circ}{n}</math> angle of rotation.

In the three examples above, the hourglass has order 2 rotation symmetry, the recycle logo has order 3 rotation symmetry, and the star has order 5 rotation symmetry.