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# Rotational Symmetry

A figure is said to have rotational symmetry if it can be rotated by an angle between 0 and 360 degrees such that the image coincides with the preimage. Put another way, a figure has rotational symmetry if you can spin it and end up with something that looks the same as it did before spinning without completing a full 360-degree rotation. Let's take a closer look at how rotational symmetry works.

## Rotational symmetry terminology

There are two key pieces of information you should know when discussing rotational symmetry. First, the angle of rotational symmetry is the smallest angle for which a figure can be rotated to coincide with itself. Second, the order of symmetry is how many times a figure coincides with itself as it rotates up to 360 degrees.

Let's look at a couple of examples to illustrate these concepts. A regular hexagon has rotational symmetry. The angle of rotation is 60 degrees and the order of rotational symmetry is 6 as illustrated by the diagram below. However, a scalene triangle has no rotational symmetry because you cannot rotate it and get the same image:

Since the scalene triangle doesn't have rotational symmetry, it also doesn't have an angle of rotation or order of rotational symmetry as the hexagon does.

## Determining rotational symmetry for regular polygons

The regular hexagon in the example above illustrates a key factor in rotational symmetry. Recall that all regular polygons have equal sides and angles and a hexagon has six sides. Since all sides and angles are equal, the hexagon will have an order of rotational symmetry equal to the number of sides (6). This remains true for all regular polygons no matter how many sides they have.

Similarly, you can calculate the angle of rotation by dividing the number of sides of a regular polygon by 360 degrees for a full turn. Returning to the example of a hexagon, we get:

$360÷6=60$

Again, this works for all regular polygons no matter how many sides they have.

## Rotational symmetry at any angle

While geometric shapes like regular polygons achieve rotational symmetry at specific angles, other shapes have rotational symmetry at any angle. Can you guess which one? If you said circle, you're right! Since circles are defined as all points within a set distance from an origin point, you can rotate it any which way and end up looking at an identical image of a circle.

However, your textbook might try to take advantage of this fact to trick you. While circles have rotational symmetry at any angle, it's no longer the case if you put a polygon into the circle or divide it in such a way that the "slices" are no longer identical. Don't automatically say infinite rotational symmetry as soon as you see a circle!

## Rotational symmetry practice questions

a. If a regular polygon has 18 sides, calculate its order of rotational symmetry.

The order of rotational symmetry for a regular polygon is equal to the number of sides. Since the polygon has 18 sides:

Order of rotational symmetry is 18

b. What is the angle of rotation for the 18-sided regular polygon above?

To find the angle of rotation for an 18-sided regular polygon, divide 360 degrees by the number of sides:

$\mathrm{Angle of rotation}=\frac{360\mathrm{degrees}}{18\mathrm{sides}}=20\mathrm{degrees}$

c. What is the order of rotational symmetry for a square? Its angle of rotation?

4 = order of rotational symmetry

$\mathrm{Angle of rotation}=\frac{360\mathrm{degrees}}{4\mathrm{sides}}=90\mathrm{degrees}$

d. Does a scalene trapezoid have rotational symmetry?

A scalene trapezoid does not have rotational symmetry because it does not look the same after a rotation of less than 360 degrees.

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